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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 1063821, 34905]*) (*NotebookOutlinePosition[ 1080269, 35491]*) (* CellTagsIndexPosition[ 1079827, 35476]*) (*WindowFrame->Normal*) Notebook[{ Cell["Laminar channel flow over a (long)wavy surface ", "Title"], Cell[TextData[{ StyleBox["This notebook has been written in ", FontSize->12], StyleBox["Mathematica ", FontSize->12, FontSlant->"Italic"], StyleBox[ "by \n\nMark J. McCready\nProfessor and Chair of Chemical Engineering\n\ University of Notre Dame\nNotre Dame IN 46556\nUSA\n\n\ Mark.J.McCready.1@nd.edu\n", FontSize->12], ButtonBox["http://www.nd.edu/~mjm/", ButtonData:>{ URL[ "http://www.nd.edu/~mjm/"], None}, ButtonStyle->"Hyperlink"], "\n\nIt is copyrighted to the extent allowed by whatever laws pertain to \ the World Wide Web and the Internet.\n\nI would hope that as a professional \ courtesy, if you use it, this notice remain visible to other users. \nThere \ is no charge for copying and dissemination \n\n", StyleBox[ "Version: 5/14/99\nMore recent versions of this notebook should be \ available at the web site:\n", FontSize->12], ButtonBox["http://www.nd.edu/~mjm/pressure.shear.fluctuations.nb", ButtonData:>{ URL[ "http://www.nd.edu/~mjm/pressure.shear.fluctuations.nb"], None}, ButtonStyle->"Hyperlink"] }], "Text", FontSize->10], Cell[CellGroupData[{ Cell["Summary", "Subtitle"], Cell[TextData[{ "This notebook attempts to provide insight into how different models for \ linear stability will predict the formation of waves and by implication, the \ transition from stratified flow to another regime. It is expected for two \ fluids, where one is much more viscous and dense, and the other has a \ Reynolds number large compared to unity, that the pressure shear variations \ over a linear wave strongly control the wave growth. See Hanratty (1983) in \ ", StyleBox["Waves on fluid interfaces, ", FontSlant->"Italic"], "p221-259", StyleBox[".", FontSlant->"Italic"], " Academic Press, Kuru, Sangalli, Uphold and McCready (1995) ", StyleBox["IJMF", FontSlant->"Italic"], ", ", StyleBox["21", FontWeight->"Bold"], ", p733-753. " }], "Text"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " aside" }], "Subsection"], Cell[TextData[{ "In ", StyleBox["Mathematica", FontSlant->"Italic"], ", it is convenient to give all expressions a \"name\". I try to pick ones \ that are consistent with what is being done (but sometimes \"temp\" is used). \ This assignment is done with an \"=\" sign. To make an equation, a \"==\" \ is used. This distinction is very useful in computer algebra and is employed \ in all of the packages with which I am familiar ." }], "Text"], Cell[CellGroupData[{ Cell["Input notation", "Subsubsection"], Cell["\<\ I have used a mix of input notation to show how it can be done. \ \ \>", "Text"], Cell["\<\ I would enter the dynamic boundary condition from a key pad as\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(dc = \(-\ \[Gamma]\)\ D[\[Eta][x, t], {x, 2}]\ - \n\t\t \(\[Rho]\_1\) \((\(-D[\[Phi]\_1[x, y, t], t]\) - U1\ D[\[Phi]\_1[x, y, t], x] - g\ \[Eta][x, t])\) + \n\t\t\t \(\[Rho]\_2\) \((\(-D[\[Phi]\_2[x, y, t], t]\) - \n\t\t\t\t\t U2\ D[\[Phi]\_2[x, y, t], x] - g\ \[Eta][x, t])\)\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(-\[Gamma]\), " ", RowBox[{ SuperscriptBox["\[Eta]", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "(", \(x, t\), ")"}]}], "-", RowBox[{\(\[Rho]\_1\), " ", RowBox[{"(", RowBox[{\(\(-g\)\ \(\[Eta](x, t)\)\), "-", RowBox[{ SubsuperscriptBox["\[Phi]", "1", TagBox[\((0, 0, 1)\), Derivative], MultilineFunction->None], "(", \(x, y, t\), ")"}], "-", RowBox[{"U1", " ", RowBox[{ SubsuperscriptBox["\[Phi]", "1", TagBox[\((1, 0, 0)\), Derivative], MultilineFunction->None], "(", \(x, y, t\), ")"}]}]}], ")"}]}], "+", RowBox[{\(\[Rho]\_2\), " ", RowBox[{"(", RowBox[{\(\(-g\)\ \(\[Eta](x, t)\)\), "-", RowBox[{ SubsuperscriptBox["\[Phi]", "2", TagBox[\((0, 0, 1)\), Derivative], MultilineFunction->None], "(", \(x, y, t\), ")"}], "-", RowBox[{"U2", " ", RowBox[{ SubsuperscriptBox["\[Phi]", "2", TagBox[\((1, 0, 0)\), Derivative], MultilineFunction->None], "(", \(x, y, t\), ")"}]}]}], ")"}]}]}], TraditionalForm]], "Output"] }, Open ]], Cell[TextData[{ "In ", StyleBox["Standard form", FontSlant->"Italic"], ", which is a little shorter, you would need the typeset window to make \ this practical. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(dc = \(-\((\[Gamma]\ \[PartialD]\_{x, 2}\[Eta][x, t])\)\) - \[Rho]\_1\ \((\(-\[PartialD]\_t \[Phi]\_1[x, y, t]\) - U1\ \[PartialD]\_x \[Phi]\_1[x, y, t] - g\ \[Eta][x, t])\) + \[Rho]\_2\ \((\(-\[PartialD]\_t \[Phi]\_2[x, y, t]\) - U2\ \[PartialD]\_x \[Phi]\_2[x, y, t] - g\ \[Eta][x, t])\)\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(-\[Gamma]\), " ", RowBox[{ SuperscriptBox["\[Eta]", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "(", \(x, t\), ")"}]}], "-", RowBox[{\(\[Rho]\_1\), " ", RowBox[{"(", RowBox[{\(\(-g\)\ \(\[Eta](x, t)\)\), "-", RowBox[{ SubsuperscriptBox["\[Phi]", "1", TagBox[\((0, 0, 1)\), Derivative], MultilineFunction->None], "(", \(x, y, t\), ")"}], "-", RowBox[{"U1", " ", RowBox[{ SubsuperscriptBox["\[Phi]", "1", TagBox[\((1, 0, 0)\), Derivative], MultilineFunction->None], "(", \(x, y, t\), ")"}]}]}], ")"}]}], "+", RowBox[{\(\[Rho]\_2\), " ", RowBox[{"(", RowBox[{\(\(-g\)\ \(\[Eta](x, t)\)\), "-", RowBox[{ SubsuperscriptBox["\[Phi]", "2", TagBox[\((0, 0, 1)\), Derivative], MultilineFunction->None], "(", \(x, y, t\), ")"}], "-", RowBox[{"U2", " ", RowBox[{ SubsuperscriptBox["\[Phi]", "2", TagBox[\((1, 0, 0)\), Derivative], MultilineFunction->None], "(", \(x, y, t\), ")"}]}]}], ")"}]}]}], TraditionalForm]], "Output"] }, Open ]], Cell[TextData[{ "Finally it may be more easily ", StyleBox["read", FontVariations->{"Underline"->True}], " in ", StyleBox["Traditional form", FontSlant->"Italic"], " as" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ FormBox[ RowBox[{"dc", "=", RowBox[{ RowBox[{"-", RowBox[{"(", RowBox[{"\[Gamma]", " ", FractionBox[\(\[PartialD]\^2\( \[Eta](x, t)\)\), \(\[PartialD]x\^2\), MultilineFunction->None]}], ")"}]}], "-", RowBox[{\(\[Rho]\_1\), " ", RowBox[{"(", RowBox[{ RowBox[{"-", FractionBox[\(\[PartialD]\[Phi]\_1[x, y, t]\), \(\[PartialD]t\), MultilineFunction->None]}], "-", RowBox[{"U1", " ", FractionBox[\(\[PartialD]\[Phi]\_1[x, y, t]\), \(\[PartialD]x\), MultilineFunction->None]}], "-", \(g\ \(\[Eta](x, t)\)\)}], ")"}]}], "+", RowBox[{\(\[Rho]\_2\), " ", RowBox[{"(", RowBox[{ RowBox[{"-", FractionBox[\(\[PartialD]\[Phi]\_2[x, y, t]\), \(\[PartialD]t\), MultilineFunction->None]}], "-", RowBox[{"U2", " ", FractionBox[\(\[PartialD]\[Phi]\_2[x, y, t]\), \(\[PartialD]x\), MultilineFunction->None]}], "-", \(g\ \(\[Eta](x, t)\)\)}], ")"}]}]}]}], TraditionalForm]], "Input"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(-\[Gamma]\), " ", RowBox[{ SuperscriptBox["\[Eta]", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "(", \(x, t\), ")"}]}], "-", RowBox[{\(\[Rho]\_1\), " ", RowBox[{"(", RowBox[{\(\(-g\)\ \(\[Eta](x, t)\)\), "-", RowBox[{ SubsuperscriptBox["\[Phi]", "1", TagBox[\((0, 0, 1)\), Derivative], MultilineFunction->None], "(", \(x, y, t\), ")"}], "-", RowBox[{"U1", " ", RowBox[{ SubsuperscriptBox["\[Phi]", "1", TagBox[\((1, 0, 0)\), Derivative], MultilineFunction->None], "(", \(x, y, t\), ")"}]}]}], ")"}]}], "+", RowBox[{\(\[Rho]\_2\), " ", RowBox[{"(", RowBox[{\(\(-g\)\ \(\[Eta](x, t)\)\), "-", RowBox[{ SubsuperscriptBox["\[Phi]", "2", TagBox[\((0, 0, 1)\), Derivative], MultilineFunction->None], "(", \(x, y, t\), ")"}], "-", RowBox[{"U2", " ", RowBox[{ SubsuperscriptBox["\[Phi]", "2", TagBox[\((1, 0, 0)\), Derivative], MultilineFunction->None], "(", \(x, y, t\), ")"}]}]}], ")"}]}]}], TraditionalForm]], "Output"] }, Open ]], Cell["\<\ Note that you can convert any \"cryptic\" input expression by \ selecting the cell, going up to the cell menu and selecting \"convert to \ traditional form\". 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1000Ool000000Woo0P000goo10000goo0`0000Aoo`00Oomoo`<0009oo`03001oogoo00<004eoo`80 015oo`03001oogoo0:=oo`8003]oo`00;7oo00D007ooOomoo`0000Eoo`03001oogoo015oo`03001o ogoo01aoo`03001oogoo02Yoo`03001oogoo00Eoo`03001oogoo051oo`80015oojL003aoo`00;7oo 00D007ooOomoo`0000Aoo`8002moo`03001oogoo00Aoo`03001oogoo02Ioo`03001oogoo00Eoo`03 001oogoo051oo`800?Aoo`00;7oo00<007ooOol00Woo100000=oo`00Ool0GWoo00<007ooOol01Goo 00<007ooOol0D7oo0P00m7oo000/Ool00`00Oomoo`32Ool2003dOol002aoo`03001oogoo0<9oo`80 0?Aoo`00;7oo00<007ooOol0`Woo0P00m7oo000/Ool00`00Oomoo`32Ool2003dOol002aoolH00?Eo o`00ogooj7oo003oOooXOol00?moonQoo`00ogooj7oo003oOooXOol00001\ \>"], "Graphics", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, Evaluatable->False, ImageSize->{487, 453}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "A number of studies have suggested that regime transitions from stratified \ to slug and/or annular flow occur when then stratified flow becomes unstable \ to long waves. The plot above shows a number of different models, most based \ on linear stability theory, to predict these flow regime transitions. They \ do not agree. Further, the integral equation models, Lin & Hanratty, Barnea, \ Crowley et al., use an integrated form of the momentum equations -- as \ opposed to the differential equations, and these results do not agree \ (compare orange versus green lines). This lack of agreement is troubling and \ there are two major issues. \n\nFirst we need to determine the inherent \ error in using differential as opposed to an integral form of the momentum \ equations. We cannot expect these two to be the same.\n\nThe second issue \ concerns the effect of turbulence. While for a laminar flow the differential \ equations can be solved for all conditions with the appropriate numerical or \ possibly analytical approach, a turbulent flow requires a model for the \ turbulence which is necessarily of somewhat limited accuracy. (Note that it \ is effectively of unknown accuracy because turbulence models have been \ \"tuned\" for various single phase flows but not for the region near the \ interface of two-phase flows). Given this question of accuracy, it might be \ reasonable to consider using an integral model for the turbulent case. \ However, it is still important to know the inherent accuracy of integral \ models.\n\nIn this notebook, we attempt to get some understanding of both of \ these issues by solving the much simpler problem of flow over a wavy surface \ and looking at the fluctuations in pressure and shear stress that result. We \ consider only the long wave region, H/\[Lambda] <<1 (amplitude/H and \ amplitude/\[Lambda] are also small). Hanratty (1983) (in ", StyleBox["Waves on fluid interfaces, ", FontSlant->"Italic"], "p221-259", StyleBox[".", FontSlant->"Italic"], " Academic Press) emphasizes how important shear and pressure variations \ are in the wave formation problem for gas-liquid systems. The pressure and \ shear stress components are be compared for different classes of problems to \ see the quantitative and qualitiative differences. The integral momentum \ model of Lin and Hanratty (1986) ", StyleBox["IJMF", FontSlant->"Italic"], " is solved along with the differential laminar equations and a \ differential model that uses a polynomial model.\n\nThe wave is chosen to be \ ", StyleBox["long", FontSlant->"Italic"], " because it is the formation and growth of the longest waves that could \ cause regime transitions. Further, the integral equations make sense only \ when the waves are long.\n\nWe hope that a comparison for laminar flow \ between the integral and differential models gives an estimate of the \ inherent accuracy of the integral model at predicing linear stability. \ Further, the comparison of the polynomial differential and laminar \ differential models should give some idea how much difference turbulence \ might be in the stability problem. " }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Description and physical situation", "Subtitle"], Cell["\<\ This notebook solves the problem of laminar flow over a wavy solid \ surface. The linearized equations are used. A long wave expansion \ perturbation technique is employed. The major point of this notebook is to \ calculate the pressure and shear stress components at the wave surface. \ These provide insight into the wave growth process. A comparison of the \ longwave perturbation and boundary-layer approximations are given. \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "The total average channel height is ", StyleBox["H", FontSlant->"Italic"], " and we nondimensionalize with it so that the top wall is at y=1. Also, \ we use the average velocity, ", Cell[BoxData[ \(TraditionalForm\`U\&_\)]], ", to nondimensionalize the velocities. " }], "Text"], Cell[GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHg"], "Graphics", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, Evaluatable->False, ImageSize->{478, 120}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}] }, Closed]], Cell[CellGroupData[{ Cell["How to navigate this notebook", "Subtitle"], Cell["\<\ It should be possible to run the \"Derivations\" section which will \ fill in answers for all of the stress and pressure variations. The \ polynomial profile section will take several minutes to run on a Mac G3. You \ could also make changes or correct my mistakes by altering these sections. \ Alternatively, you can just use my answers by clicking on the \"Results\" \ section. \ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Preview of conclusions", "Subtitle"], Cell["\<\ A comparison of the laminar differential and integral models \ suggests that the first terms match exactly which for the stability problem \ control the wave speed. However there is about a 20% difference between the \ pressure at second order and the integral model does not produce a second \ term in the shear stress. The second term controls the wave growth and while \ the shear stress seems to be less important than pressure for wave growth, we \ expect at least this 20% difference to show up. Thus we must expect a \ significant difference between differential and integral stability analyses. The comparison of the laminar and polynomial differential models shows \ consistent significant differences. Thus we can expect a difference in the \ stability boundaries. This is, of course, expected. The problem is that we \ can't tell exactly how to normalize the effects. \ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Derivations of pressure and shear variations using all of the \ models\ \>", "Subtitle"], Cell[CellGroupData[{ Cell[TextData[{ "Pressure and shear stress variations from the linearized ", StyleBox["differential", FontSlant->"Italic"], " equation for a ", StyleBox["laminar", FontVariations->{"Underline"->True}], " flow. " }], "Section"], Cell[CellGroupData[{ Cell["Base state channel flow", "Subsection"], Cell[TextData[{ "We want to solve the problem of a laminar flow in a channel in which one \ wall is wavy. We will locate the origin at the average location of the wavy \ wall and the channel top at ", StyleBox["y", FontSlant->"Italic"], "=", StyleBox["h", FontSlant->"Italic"], ". The average velocity profile will be parabolic, and we will be doing a \ perturbation solution, so first find the parabolic profile solution. The \ governing equation is 0 = -", Cell[BoxData[ \(\(\(1\/\[Rho]\) \[PartialD]p\/\[PartialD]x + \ \[Nu] \[PartialD]\^2 u\/\((\[PartialD]y)\)\^2\ \)\)]], " with no slip boundary conditions at ", StyleBox["y", FontSlant->"Italic"], " = 0 and ", StyleBox["h", FontSlant->"Italic"], ". \n\nIf we nondimensionalize with a length, ", StyleBox["h", FontSlant->"Italic"], " and the average velocity, ", StyleBox["U", FontSlant->"Italic"], ", we could solve this directly and get the parabolic profile as ", Cell[BoxData[ \(u \((y)\)\ = \ 6 y\ + \ 6\ \(y\ \)\^2\)]], " for u(y) nondimensionalized with the average velocity. \n\nHowever, lets \ solve this using the stream function, ", Cell[BoxData[ \(\(\[Psi]\_0\) \(\((y)\) . \)\)]], "\n\nFor ", Cell[BoxData[ \(\(\[Psi]\_0\) \((y)\)\)]], " defined using:\n\n\tu(y) = ", Cell[BoxData[ \(\(\[PartialD]\[Psi]\_0 \((y)\)\)\/\[PartialD]y\)]], " and v(y) = - ", Cell[BoxData[ \(\(\[PartialD]\[Psi]\_0 \((y)\)\)\/\(\[PartialD]x\ \)\)]], "\n\t\nwhere ", Cell[BoxData[ \(u \((y)\)\)]], " is the flow direction velocity, we would substitute for u(y) with ", Cell[BoxData[ \(\(\[PartialD]\[Psi]\_0 \((y)\)\)\/\[PartialD]y\)]], " and then take a derivative with y to get rid of the pressure. The result \ is: ", Cell[BoxData[ \(\(\[PartialD]\^4\ \[Psi]\_0 \((y)\)\)\/\[PartialD]y\^4\)]], " = 0.\n\nThe form of ", Cell[BoxData[ \(\[Psi]\_0\)]], "(y) is clearly a cubic polynomial." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(\[Psi]0\ = a3\ y\^3 + a2\ y\^2 + a1\ y + a0\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm\`a3\ y\^3 + a2\ y\^2 + a1\ y + a0\)], "Output"], Cell[TextData[{ "The value of ", Cell[BoxData[ \(a0\)]], " is arbitrary because the governing equations are for the velocity, not \ the stream function, so they always involve derivatives of ", Cell[BoxData[ \(\[PartialD]\[Psi]\_0 \((y\)\)]], "). \n\nLet's fit the boundary conditions." }], "Text"], Cell["At the bottom (wavy) wall, u(0)=0", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(bc1 = \((\[PartialD]\_y \[Psi]0 /. y \[Rule] 0)\)\ == 0\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm\`a1 == 0\)], "Output"] }, Open ]], Cell["At the top wall, u(1)=0", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(bc2 = \((\[PartialD]\_y \[Psi]0 /. y \[Rule] 1)\)\ == 0\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm\`a1 + 2\ a2 + 3\ a3 == 0\)], "Output"] }, Open ]], Cell["Lets solve these,", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ans1 = Solve[{bc1, bc2}, {a1, a2}]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm\`{{a1 \[Rule] 0, a2 \[Rule] \(-\(\(3\ a3\)\/2\)\)}}\)], "Output"] }, Open ]], Cell["We can substitute:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(psi0temp1 = \[Psi]0 /. ans1[\([1]\)]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`a3\ y\^3 - \(3\ a3\ y\^2\)\/2 + a0\)], "Output"] }, Open ]], Cell[TextData[{ "We might as well choose ", Cell[BoxData[ \(\(\[Psi]\_0\) \((0)\)\)]], " = 0 " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(psi0temp2 = psi0temp1 /. a0 -> 0\)], "Input"], Cell[BoxData[ \(TraditionalForm\`a3\ y\^3 - \(3\ a3\ y\^2\)\/2\)], "Output"] }, Open ]], Cell["\<\ To get the last constant, we need to make the average flow rate \ match the real flow rate which is unity for the nondimensional problem.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(temp1 = Integrate[D[psi0temp2, y], {y, 0, 1}]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`\(-\(a3\/2\)\)\)], "Output"] }, Open ]], Cell["We then can get a3 by", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(temp2 = Solve[temp1 == 1, a3]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm\`{{a3 \[Rule] \(-2\)}}\)], "Output"] }, Open ]] }, Closed]], Cell["Thus the stream function for the average problem is", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(psi0ans = psi0temp2 /. temp2[\([1]\)]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`3\ y\^2 - 2\ y\^3\)], "Output"] }, Open ]], Cell["Check to make sure that the velocity field is correct", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(uave = D[psi0ans, y]\)], "Input", 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Cell[CellGroupData[{ Cell["We also need the interfacial shear", "Subsubsection"], Cell[TextData[{ "The shear stress at the wall is given by, 1/R (\[PartialD]v/\[PartialD]x+ \ \[PartialD]u/\[PartialD]y). We will need to expand this aroung y=0, (domain \ perturbation) because the wall is not at 0, it is a y= \[Eta] and we don't \ want to have to evaluate the boundary condition there all of the time. \n\n\ This should be\n\n\[Tau](0) = 1/", "R", " (", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\^2\)]], " \[Phi]+\[PartialD]\[Phi]/\[PartialD]y + \[PartialD]", Cell[BoxData[ \(TraditionalForm\`U\&_\)]], "/\[PartialD]y) @ y=0 " }], "Text"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Here we solve the 0 order problem", "Subsection"], Cell[CellGroupData[{ Cell[TextData["Here is the first perturbation term:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(eq0\)], "Input", AspectRatioFixed->True], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{"\[ImaginaryI]", " ", RowBox[{ SubsuperscriptBox["\[Phi]", "0", TagBox[\((4)\), Derivative], MultilineFunction->None], "(", "y", ")"}]}], "R"], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(eq0t1 = DSolve[eq0 == 0, \[Phi]\_0[y], y]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{"{", RowBox[{\(\(\[Phi]\_0\)(y)\), "\[Rule]", RowBox[{ RowBox[{ SubscriptBox[ TagBox["c", C], "4"], " ", \(y\^3\)}], "+", RowBox[{ SubscriptBox[ TagBox["c", C], "3"], " ", \(y\^2\)}], "+", RowBox[{ SubscriptBox[ TagBox["c", C], "2"], " ", "y"}], "+", SubscriptBox[ TagBox["c", C], "1"]}]}], "}"}], "}"}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(eq0t2 = \[Phi]\_0[y] /. eq0t1\[LeftDoubleBracket]1\[RightDoubleBracket]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox[ TagBox["c", C], "4"], " ", \(y\^3\)}], "+", RowBox[{ SubscriptBox[ TagBox["c", C], "3"], " ", \(y\^2\)}], "+", RowBox[{ SubscriptBox[ TagBox["c", C], "2"], " ", "y"}], "+", SubscriptBox[ TagBox["c", C], "1"]}], TraditionalForm]], "Output"] }, Open ]], Cell["\<\ The first boundary condition is that the top wall normal velocity \ is 0\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(bc01 = eq0t2 /. {y \[Rule] 1}\)], "Input", AspectRatioFixed->True], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox[ TagBox["c", C], "1"], "+", SubscriptBox[ TagBox["c", C], "2"], "+", SubscriptBox[ TagBox["c", C], "3"], "+", SubscriptBox[ TagBox["c", C], "4"]}], TraditionalForm]], "Output"] }, Open ]], Cell["\<\ The next boundary condition is that the top wall tangential \ velocity is 0\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(bc02 = \[PartialD]\_y eq0t2 /. y \[Rule] 1\)], "Input", AspectRatioFixed->True], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox[ TagBox["c", C], "2"], "+", RowBox[{"2", " ", SubscriptBox[ TagBox["c", C], "3"]}], "+", RowBox[{"3", " ", SubscriptBox[ TagBox["c", C], "4"]}]}], TraditionalForm]], "Output"] }, Open ]], Cell["At the wavy wall, the normal velocity is 0", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(bc03 = eq0t2 /. y \[Rule] 0\)], "Input", AspectRatioFixed->True], Cell[BoxData[ FormBox[ SubscriptBox[ TagBox["c", C], "1"], TraditionalForm]], "Output"] }, Open ]], Cell[TextData[ "At the wavy wall, the tangential velocity is 0. Note that the wall is not \ always at 0 so we need to use domain perturbation to fix this. It involves \ expanding the tangential velocity around 0.\n\nThe result is that \ u(\[Eta])=0= u'(0) + \[PartialD]", Cell[BoxData[ \(TraditionalForm\`\(\(U\&_\)(0)\)/\[PartialD]y\)]]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(bc04a = \((\[PartialD]\_y eq0t2 + \[PartialD]\_{y, 1}uave)\)\)], "Input",\ AspectRatioFixed->True], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"3", " ", SubscriptBox[ TagBox["c", C], "4"], " ", \(y\^2\)}], "+", RowBox[{"2", " ", SubscriptBox[ TagBox["c", C], "3"], " ", "y"}], "-", \(12\ y\), "+", SubscriptBox[ TagBox["c", C], "2"], "+", "6"}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(bc04 = bc04a /. y -> 0\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox[ TagBox["c", C], "2"], "+", "6"}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(consts0 = Solve[{bc01 == 0, bc02 == 0, bc03 == 0, bc04 == 0}, {C[1], C[2], C[3], C[4]}]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{ SubscriptBox[ TagBox["c", C], "1"], "\[Rule]", "0"}], ",", RowBox[{ SubscriptBox[ TagBox["c", C], "2"], "\[Rule]", \(-6\)}], ",", RowBox[{ SubscriptBox[ TagBox["c", C], "3"], "\[Rule]", "12"}], ",", RowBox[{ SubscriptBox[ TagBox["c", C], "4"], "\[Rule]", \(-6\)}]}], "}"}], "}"}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(phi0 = eq0t2 /. consts0[\([1]\)]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`\(-6\)\ y\^3 + 12\ y\^2 - 6\ y\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Calculate the 0 order pressure", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"pans01", "=", RowBox[{"peq0", "/.", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox[\(\[Phi]\_0\), "\[Prime]", MultilineFunction->None], "[", "y", "]"}], "\[Rule]", \(\[PartialD]\_y phi0\)}], ",", \(\[Phi]\_0[y] \[Rule] phi0\), ",", RowBox[{ RowBox[{ SuperscriptBox[\(\[Phi]\_0\), TagBox[\((3)\), Derivative], MultilineFunction->None], "[", "y", "]"}], "\[Rule]", \(\[PartialD]\_{y, 3}phi0\)}]}], "}"}]}]}]], "Input", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm\`\(36\ \[ImaginaryI]\)\/R\)], "Output"] }, Open ]], Cell[TextData[{ "We recall that the first term has ", Cell[BoxData[ \(TraditionalForm\`\(\ \ \[Alpha]\^\(-1\)\)\)]], "power so we write" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(pressure0 = pans01/\[Alpha]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`\(36\ \[ImaginaryI]\)\/\(R\ \[Alpha]\)\)], "Output"] }, Open ]] }, Open ]], Cell["Calculate the 0 order shear stress", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(stress0 = \((1/R \((D[phi0, {y, 2}] + \[PartialD]\_{y, 2}uave)\))\) /. y -> 0\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm\`12\/R\)], "Output"] }, Open ]] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Here we solve the first order problem", "Subsection"], Cell[CellGroupData[{ Cell["Here is the next perturbation term:", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(eq1\)], "Input", AspectRatioFixed->True], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(-6\), " ", RowBox[{ SubsuperscriptBox["\[Phi]", "0", "\[DoublePrime]", MultilineFunction->None], "(", "y", ")"}], " ", \(y\^2\)}], "+", RowBox[{"6", " ", RowBox[{ SubsuperscriptBox["\[Phi]", "0", "\[DoublePrime]", MultilineFunction->None], "(", "y", ")"}], " ", "y"}], "+", \(12\ \(\(\[Phi]\_0\)(y)\)\), "+", FractionBox[ RowBox[{"\[ImaginaryI]", " ", RowBox[{ SubsuperscriptBox["\[Phi]", "1", TagBox[\((4)\), Derivative], MultilineFunction->None], "(", "y", ")"}]}], "R"]}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(eq1t1 = eq1 /. {\[Phi]\_0[y] -> phi0, \(\(\[Phi]\_0'\)'\)[y] -> D[phi0, {y, 2}]} \)], "Input"], Cell[BoxData[ FormBox[ RowBox[{ \(\(-6\)\ \((24 - 36\ y)\)\ y\^2\), "+", \(6\ \((24 - 36\ y)\)\ y\), "+", \(12\ \((\(-6\)\ y\^3 + 12\ y\^2 - 6\ y)\)\), "+", FractionBox[ RowBox[{"\[ImaginaryI]", " ", RowBox[{ SubsuperscriptBox["\[Phi]", "1", TagBox[\((4)\), Derivative], MultilineFunction->None], "(", "y", ")"}]}], "R"]}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(eq1t2 = DSolve[eq1t1 == 0, \[Phi]\_1[y], y]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{"{", RowBox[{\(\(\[Phi]\_1\)(y)\), "\[Rule]", RowBox[{ \(6\/35\ \[ImaginaryI]\ R\ y\^7\), "-", \(3\/5\ \[ImaginaryI]\ R\ y\^6\), "+", \(3\/5\ \[ImaginaryI]\ R\ y\^5\), "+", RowBox[{ SubscriptBox[ TagBox["c", C], "4"], " ", \(y\^3\)}], "+", RowBox[{ SubscriptBox[ TagBox["c", C], "3"], " ", \(y\^2\)}], "+", RowBox[{ SubscriptBox[ TagBox["c", C], "2"], " ", "y"}], "+", SubscriptBox[ TagBox["c", C], "1"]}]}], "}"}], "}"}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(eq1t3 = \[Phi]\_1[y] /. eq1t2\[LeftDoubleBracket]1\[RightDoubleBracket]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ FormBox[ RowBox[{ \(6\/35\ \[ImaginaryI]\ R\ y\^7\), "-", \(3\/5\ \[ImaginaryI]\ R\ y\^6\), "+", \(3\/5\ \[ImaginaryI]\ R\ y\^5\), "+", RowBox[{ SubscriptBox[ TagBox["c", C], "4"], " ", \(y\^3\)}], "+", RowBox[{ SubscriptBox[ TagBox["c", C], "3"], " ", \(y\^2\)}], "+", RowBox[{ SubscriptBox[ TagBox["c", C], "2"], " ", "y"}], "+", SubscriptBox[ TagBox["c", C], "1"]}], TraditionalForm]], "Output"] }, Open ]], Cell["\<\ The first boundary condition is that the top wall normal velocity \ is 0\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(bc11 = eq1t3 /. {y \[Rule] 1}\)], "Input", AspectRatioFixed->True], Cell[BoxData[ FormBox[ RowBox[{\(\(6\ \[ImaginaryI]\ R\)\/35\), "+", SubscriptBox[ TagBox["c", C], "1"], "+", SubscriptBox[ TagBox["c", C], "2"], "+", SubscriptBox[ TagBox["c", C], "3"], "+", SubscriptBox[ TagBox["c", C], "4"]}], TraditionalForm]], "Output"] }, Open ]], Cell["\<\ The next boundary condition is that the top wall tangential \ velocity is 0\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(bc12 = \[PartialD]\_y eq1t3 /. y \[Rule] 1\)], "Input", AspectRatioFixed->True], Cell[BoxData[ FormBox[ RowBox[{\(\(3\ \[ImaginaryI]\ R\)\/5\), "+", SubscriptBox[ TagBox["c", C], "2"], "+", RowBox[{"2", " ", SubscriptBox[ TagBox["c", C], "3"]}], "+", RowBox[{"3", " ", SubscriptBox[ TagBox["c", C], "4"]}]}], TraditionalForm]], "Output"] }, Open ]], Cell["At the wavy wall, the normal velocity is 0", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(bc13 = eq1t3 /. y \[Rule] 0\)], "Input", AspectRatioFixed->True], Cell[BoxData[ FormBox[ SubscriptBox[ TagBox["c", C], "1"], TraditionalForm]], "Output"] }, Open ]], Cell[TextData[{ "At the wavy wall, the tangential velocity is 0.\nAt first order we have \ that \[PartialD]", Cell[BoxData[ \(TraditionalForm\`\[Phi]\_1\)]], "/\[PartialD]y = 0. The average velocity is only in the lowest order \ \[Alpha] equation. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(bc14 = \((\[PartialD]\_y eq1t3)\) /. y -> 0\)], "Input", AspectRatioFixed->True], Cell[BoxData[ FormBox[ SubscriptBox[ TagBox["c", C], "2"], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(consts1 = Solve[{bc11 == 0, bc12 == 0, bc13 == 0, bc14 == 0}, {C[1], C[2], C[3], C[4]}]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{ SubscriptBox[ TagBox["c", C], "1"], "\[Rule]", "0"}], ",", RowBox[{ SubscriptBox[ TagBox["c", C], "2"], "\[Rule]", "0"}], ",", RowBox[{ SubscriptBox[ TagBox["c", C], "3"], "\[Rule]", \(\(3\ \[ImaginaryI]\ R\)\/35\)}], ",", RowBox[{ SubscriptBox[ TagBox["c", C], "4"], "\[Rule]", \(-\(\(9\ \[ImaginaryI]\ R\)\/35\)\)}]}], "}"}], "}"}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(phi1 = eq1t3 /. consts1[\([1]\)]\)], "Input"], Cell[BoxData[ \(TraditionalForm \`6\/35\ \[ImaginaryI]\ R\ y\^7 - 3\/5\ \[ImaginaryI]\ R\ y\^6 + 3\/5\ \[ImaginaryI]\ R\ y\^5 - 9\/35\ \[ImaginaryI]\ R\ y\^3 + 3\/35\ \[ImaginaryI]\ R\ y\^2\)], "Output"] }, Open ]], Cell["Calculate the first order pressure at the wall", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(peq1\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"6", " ", RowBox[{ SubsuperscriptBox["\[Phi]", "0", "\[Prime]", MultilineFunction->None], "(", "y", ")"}], " ", \(y\^2\)}], "-", RowBox[{"6", " ", RowBox[{ SubsuperscriptBox["\[Phi]", "0", "\[Prime]", MultilineFunction->None], "(", "y", ")"}], " ", "y"}], "+", RowBox[{\(\(\[Phi]\_0\)(y)\), " ", RowBox[{ SuperscriptBox["u", "\[Prime]", MultilineFunction->None], "(", "y", ")"}]}], "-", FractionBox[ RowBox[{"\[ImaginaryI]", " ", RowBox[{ SubsuperscriptBox["\[Phi]", "1", TagBox[\((3)\), Derivative], MultilineFunction->None], "(", "y", ")"}]}], "R"]}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"pans11", "=", RowBox[{"peq1", "/.", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox[\(\[Phi]\_0\), "\[Prime]", MultilineFunction->None], "[", "y", "]"}], "\[Rule]", \(\[PartialD]\_y phi0\)}], ",", \(\[Phi]\_0[y] \[Rule] phi0\), ",", RowBox[{ RowBox[{ SuperscriptBox[\(\[Phi]\_0\), TagBox[\((3)\), Derivative], MultilineFunction->None], "[", "y", "]"}], "\[Rule]", \(\[PartialD]\_{y, 3}phi0\)}], ",", "\n", "\t\t", RowBox[{ RowBox[{ SuperscriptBox[\(\[Phi]\_1\), "\[Prime]", MultilineFunction->None], "[", "y", "]"}], "\[Rule]", \(\[PartialD]\_y phi1\)}], ",", \(\[Phi]\_1[y] \[Rule] phi1\), ",", RowBox[{ RowBox[{ SuperscriptBox[\(\[Phi]\_1\), TagBox[\((3)\), Derivative], MultilineFunction->None], "[", "y", "]"}], "\[Rule]", \(\[PartialD]\_{y, 3}phi1\)}]}], "}"}]}]}]], "Input", AspectRatioFixed->True], Cell[BoxData[ FormBox[ RowBox[{ \(6\ \((\(-18\)\ y\^2 + 24\ y - 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We include a perturbation parameter, \[Epsilon], which \ is giving an idea how large the waves are. We need \[Epsilon] to be small \ compared to ", StyleBox["H", FontSlant->"Italic"], ", the channel height, to make the linearization assumption be valid. " }], "Text"], Cell[CellGroupData[{ Cell["One dimensional mass balance ", "Subsubsection"], Cell["\<\ The mass balance relates changes in channel height to velocity \ changes for an incompresible fluid.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[PartialD]\_x\((u[x]\ h[x])\) == 0\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{\(u(x)\), " ", RowBox[{ SuperscriptBox["h", "\[Prime]", MultilineFunction->None], "(", "x", ")"}]}], "+", RowBox[{\(h(x)\), " ", RowBox[{ SuperscriptBox["u", "\[Prime]", MultilineFunction->None], "(", "x", ")"}]}]}], "==", "0"}], TraditionalForm]], "Output"] }, Open ]], Cell["\<\ We expand this in terms of the average and fluctuating \ components.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(mass1 = \[PartialD]\_x\(( \((U\&_ + \[Epsilon]\ u\&^\ \ Exp[I\ k\ x])\)\ \((H\&_ - \[Epsilon]\ h\&^\ \ Exp[I\ k\ x])\))\)\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm \`\[ImaginaryI]\ \[ExponentialE]\^\(\[ImaginaryI]\ k\ x\)\ k\ \[Epsilon]\ \((H\&_ - 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\[ImaginaryI]\ k\ U\&_\ \(h\&^\)\)], "Output"] }, Open ]], Cell["The substitution relation is", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(massans = Solve[mass3 == 0, u\&^]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm\`{{\(u\&^\) \[Rule] \(U\&_\ \(h\&^\)\)\/H\&_}}\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Integral momentum equation", "Subsubsection"], Cell["\<\ Here is the integral momentum equation in a form that will be \ expanded to get the average and perturbation equations. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(momeq = \[CapitalGamma]\ \[PartialD]\_x\(( \((H\&_ - \[Epsilon]\ h[x])\)\ \((U\&_ + \[Epsilon]\ u[x])\)\^2) \) + \n\t\(\((H\&_ - \[Epsilon]\ h[x])\)\ \((dpdx + \[Epsilon]\ \[PartialD]\_x p[x] + g\ \[Rho]\ \[PartialD]\_x\((H\&_ - \[Epsilon]\ h[x])\)\ Sin[\[Theta]])\)\)\/\[Rho] + \(\[Tau]\_b0 + \[Tau]\_w0 + \[Epsilon]\ \[Tau]\_b[x] + \[Epsilon]\ \[Tau]\_w[x]\)\/\[Rho]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ FormBox[ RowBox[{ \(\(\[Tau]\_b0 + \[Tau]\_w0 + \[Epsilon]\ \(\(\[Tau]\_b\)(x)\) + \[Epsilon]\ \(\(\[Tau]\_w\)(x)\)\)\/\[Rho]\), "+", FractionBox[ RowBox[{\((H\&_ - \[Epsilon]\ \(h(x)\))\), " ", RowBox[{"(", RowBox[{"dpdx", "-", RowBox[{ "g", " ", "\[Epsilon]", " ", "\[Rho]", " ", \(sin(\[Theta])\), " ", RowBox[{ SuperscriptBox["h", "\[Prime]", MultilineFunction->None], "(", "x", ")"}]}], "+", RowBox[{"\[Epsilon]", " ", RowBox[{ SuperscriptBox["p", "\[Prime]", MultilineFunction->None], "(", "x", ")"}]}]}], ")"}]}], "\[Rho]"], "+", RowBox[{"\[CapitalGamma]", " ", RowBox[{"(", RowBox[{ RowBox[{ "2", " ", "\[Epsilon]", " ", \((H\&_ - \[Epsilon]\ \(h(x)\))\), " ", \((U\&_ + \[Epsilon]\ \(u(x)\))\), " ", RowBox[{ SuperscriptBox["u", "\[Prime]", MultilineFunction->None], "(", "x", ")"}]}], "-", RowBox[{ "\[Epsilon]", " ", \(\((U\&_ + \[Epsilon]\ \(u(x)\))\)\^2\), " ", RowBox[{ SuperscriptBox["h", "\[Prime]", MultilineFunction->None], "(", "x", ")"}]}]}], ")"}]}]}], TraditionalForm]], "Output"] }, Open ]], Cell[TextData[ "This equation is expanded and the individual powers of \[Epsilon] are \ collected."], "Text"], Cell["ex1=Expand[momeq];", "Input", AspectRatioFixed->True], Cell[TextData["eqs1=Collect[ex1,\[Epsilon]];"], "Input", AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["First solve the basestate equation", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData["basestate1=Coefficient[eqs1,\[Epsilon],0]"], "Input", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm \`\(dpdx\ H\&_\)\/\[Rho] + \[Tau]\_b0\/\[Rho] + \[Tau]\_w0\/\[Rho]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(basestate2 = basestate1 /. {\[Tau]\_w0 \[Rule] \[Tau]\_b0}\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm\`\(dpdx\ H\&_\)\/\[Rho] + \(2\ \[Tau]\_b0\)\/\[Rho]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(wallshearstress = Solve[basestate2 == 0, \[Tau]\_b0]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm\`{{\[Tau]\_b0 \[Rule] \(-\(\(dpdx\ H\&_\)\/2\)\)}}\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Calculation of the shear stress", "Subsubsection"], Cell["\<\ We will need to eliminate the shear fluctuations in terms of \ average shear and the heights. The way that this can be done is to assume \ that the spatial variations in the flow is slow enough that the average \ relation for the friction factor (or wall shear stress) holds at any location \ along the wave. This is implemented by expanding the average relation in \ terms of a average and a perturbation equation. Start with the average relation for shear stress and Reynolds number. \ \>", "Text"], Cell[BoxData[ \(taueq1 = \[Tau] - \ 6\ \(\(\[Mu]\ /\[Rho]\)/U\)/H\)], "Input"], Cell[BoxData[ \(TraditionalForm\`\[Tau] - \(6\ \[Mu]\)\/\(H\ U\ \[Rho]\)\)], "Output"], Cell["Now expand. ", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(taueq2 = taueq1 /. {\[Tau] -> \[Tau]\&_ + \ \[Epsilon]\ \[Tau]\&^, \ H -> H\&_ + \ \[Epsilon]\ h\&^, \ U -> U\&_\ + \n\t\t\t\t\t\[Epsilon]\ u\&^}\)], "Input"], Cell[BoxData[ \(TraditionalForm \`\(-\(\(6\ \[Mu]\)\/\(\[Rho]\ \((H\&_ + \[Epsilon]\ \(h\&^\))\)\ \((U\&_ + \[Epsilon]\ \(u\&^\))\)\)\)\) + \[Tau]\&_ + \[Epsilon]\ \(\[Tau]\&^\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(taueq3 = Series[taueq2, {\[Epsilon], 0, 2}]\)], "Input"], Cell[BoxData[ FormBox[ InterpretationBox[ RowBox[{ \((\[Tau]\&_ - \(6\ \[Mu]\)\/\(\[Rho]\ H\&_\ U\&_\))\), "+", \(\((\(\[Tau]\&^\) - \(6\ \[Mu]\ \((\(-\(\(h\&^\)\/\(H\&_\^2\ U\&_\)\)\) - \(u\&^\)\/\(H\&_\ U\&_\^2\))\)\)\/\[Rho])\)\ \[Epsilon]\), "-", \(\(6\ \(( \[Mu]\ \(( \(h\&^\)\^2\/\(H\&_\^3\ U\&_\) + \(\(u\&^\)\ \(h\&^\)\)\/\(H\&_\^2\ U\&_\^2\) + \(u\&^\)\^2\/\(H\&_\ U\&_\^3\))\))\)\ \[Epsilon]\^2\)\/\[Rho]\), "+", InterpretationBox[\(O(\[Epsilon]\^3)\), SeriesData[ \[Epsilon], 0, {}, 0, 3, 1]]}], SeriesData[ \[Epsilon], 0, { Plus[ Times[ -6, \[Mu], Power[ \[Rho], -1], Power[ OverBar[ H], -1], Power[ OverBar[ U], -1]], OverBar[ \[Tau]]], Plus[ Times[ -6, \[Mu], Power[ \[Rho], -1], Plus[ Times[ -1, Power[ OverBar[ H], -2], Power[ OverBar[ U], -1], OverHat[ h]], Times[ -1, Power[ OverBar[ H], -1], Power[ OverBar[ U], -2], OverHat[ u]]]], OverHat[ \[Tau]]], Times[ -6, \[Mu], Power[ \[Rho], -1], Plus[ Times[ Power[ OverBar[ H], -3], Power[ OverBar[ U], -1], Power[ OverHat[ h], 2]], Times[ Power[ OverBar[ H], -2], Power[ OverBar[ U], -2], OverHat[ h], OverHat[ u]], Times[ Power[ OverBar[ H], -1], Power[ OverBar[ U], -3], Power[ OverHat[ u], 2]]]]}, 0, 3, 1]], TraditionalForm]], "Output"] }, Open ]], Cell["We first recover the average equation.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(averagetau = Coefficient[taueq3, \[Epsilon], 0]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`\[Tau]\&_ - \(6\ \[Mu]\)\/\(\[Rho]\ H\&_\ U\&_\)\)], "Output"] }, Open ]], Cell["We next obtain the first order term", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(tau10 = \ Coefficient[taueq3, \[Epsilon], 1]\)], "Input"], Cell[BoxData[ \(TraditionalForm \`\(\[Tau]\&^\) - \(6\ \[Mu]\ \((\(-\(\(h\&^\)\/\(H\&_\^2\ U\&_\)\)\) - \(u\&^\)\/\(H\&_\ U\&_\^2\))\)\)\/\[Rho]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(tau11 = tau10 /. massans[\([1]\)]\)], "Input"], Cell[BoxData[ \(TraditionalForm \`\(12\ \[Mu]\ \(h\&^\)\)\/\(\[Rho]\ H\&_\^2\ U\&_\) + \(\[Tau]\&^\)\)], "Output"] }, Open ]], Cell["\<\ This relation for the perturbation shear stress is the same for \ both the top and bottom walls. Note that the perturbation shear stress is \ twice the average shear stress.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(tauans = FullSimplify[Solve[tau11 == 0, \ \[Tau]\&^]]\)], "Input"], Cell[BoxData[ \(TraditionalForm \`{{\(\[Tau]\&^\) \[Rule] \(-\(\(12\ \[Mu]\ \(h\&^\)\)\/\(\[Rho]\ H\&_\^2\ U\&_\)\)\)}}\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Now solve the linear perturbation problem", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData["eq1=Coefficient[eqs1,\[Epsilon],1]"], "Input", AspectRatioFixed->True], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(-\[CapitalGamma]\), " ", RowBox[{ SuperscriptBox["h", "\[Prime]", MultilineFunction->None], "(", "x", ")"}], " ", \(U\&_\^2\)}], "+", RowBox[{"2", " ", "\[CapitalGamma]", " ", \(H\&_\), " ", RowBox[{ SuperscriptBox["u", "\[Prime]", MultilineFunction->None], "(", "x", ")"}], " ", \(U\&_\)}], "-", \(\(dpdx\ \(h(x)\)\)\/\[Rho]\), "+", \(\(\(\[Tau]\_b\)(x)\)\/\[Rho]\), "+", \(\(\(\[Tau]\_w\)(x)\)\/\[Rho]\), "-", RowBox[{"g", " ", \(H\&_\), " ", \(sin(\[Theta])\), " ", RowBox[{ SuperscriptBox["h", "\[Prime]", MultilineFunction->None], "(", "x", ")"}]}], "+", FractionBox[ RowBox[{\(H\&_\), " ", RowBox[{ SuperscriptBox["p", "\[Prime]", MultilineFunction->None], "(", "x", ")"}]}], "\[Rho]"]}], TraditionalForm]], "Output"] }, Open ]], Cell["\<\ We now use a normal mode expansion, that is, assume that the shape \ of the surface, h[x], can be written as one (or a whole sum of) sinuous \ modes. Of course, since the equations are linear, these modes will be \ independent. We use complex exponentials since they will cancel out easier.\ \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"eq2", "=", RowBox[{"eq1", "/.", RowBox[{"{", RowBox[{ \(h[x] \[Rule] h\&^\ Exp[I\ k\ x]\), ",", \(\[Tau]\_w[x] \[Rule] \[Tau]\&^\_w\ \ Exp[I\ k\ x]\), ",", \(\[Tau]\_b[x] \[Rule] \[Tau]\&^\_b\ \ Exp[I\ k\ x]\), ",", RowBox[{ RowBox[{ SuperscriptBox["h", "\[Prime]", MultilineFunction->None], "[", "x", "]"}], "\[Rule]", \(\[PartialD]\_x\((h\&^\ Exp[I\ k\ x])\)\)}], ",", RowBox[{ RowBox[{ SuperscriptBox["p", "\[Prime]", MultilineFunction->None], "[", "x", "]"}], "\[Rule]", \(\[PartialD]\_x\((p\&^\ Exp[I\ k\ x])\)\)}], ",", RowBox[{ RowBox[{ SuperscriptBox["u", "\[Prime]", MultilineFunction->None], "[", "x", "]"}], "\[Rule]", \(\[PartialD]\_x\((u\&^\ Exp[I\ k\ x])\)\)}]}], "}"}]}]}]], "Input", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm \`\(-\[ImaginaryI]\)\ \[ExponentialE]\^\(\[ImaginaryI]\ k\ x\)\ k\ \[CapitalGamma]\ \(h\&^\)\ U\&_\^2 + 2\ \[ImaginaryI]\ \[ExponentialE]\^\(\[ImaginaryI]\ k\ x\)\ k\ \[CapitalGamma]\ H\&_\ \(u\&^\)\ U\&_ - \(dpdx\ \[ExponentialE]\^\(\[ImaginaryI]\ k\ x\)\ \(h\&^\)\)\/\[Rho] + \(\[ImaginaryI]\ \[ExponentialE]\^\(\[ImaginaryI]\ k\ x\)\ k\ H\&_\ \(p\&^\)\)\/\[Rho] - \[ImaginaryI]\ \[ExponentialE]\^\(\[ImaginaryI]\ k\ x\)\ g\ k\ H\&_\ \(h\&^\)\ \(sin(\[Theta])\) + \(\[ExponentialE]\^\(\[ImaginaryI]\ k\ x\)\ \(\[Tau]\&^\)\_b\)\/\[Rho] + \(\[ExponentialE]\^\(\[ImaginaryI]\ k\ x\)\ \(\[Tau]\&^\)\_w\)\/\[Rho]\)], "Output"] }, Open ]], Cell["\<\ It is worth mentioning at this point, that this is no longer a \ differential equation. The restriction of the solution to a periodic domain \ is enough to remove the first derivatives in x. The higher order \ x-derivatives were removed in the \"averaging\" (i.e., when we decided to \ just do long waves and when we integrated in the normal direction. \ \>", "Text"], Cell[CellGroupData[{ Cell["eq3=Expand[eq2/Exp[I k x]]", "Input", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm \`\(-\[ImaginaryI]\)\ k\ \[CapitalGamma]\ \(h\&^\)\ U\&_\^2 + 2\ \[ImaginaryI]\ k\ \[CapitalGamma]\ H\&_\ \(u\&^\)\ U\&_ - \(dpdx\ \(h\&^\)\)\/\[Rho] + \(\[ImaginaryI]\ k\ H\&_\ \(p\&^\)\)\/\[Rho] - \[ImaginaryI]\ g\ k\ H\&_\ \(h\&^\)\ \(sin(\[Theta])\) + \(\[Tau]\&^\)\_b\/\[Rho] + \(\[Tau]\&^\)\_w\/\[Rho]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(pans = Solve[eq3 == 0, p\&^]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm \`{{\(p\&^\) \[Rule] \(-\(\(\(-k\)\ \[CapitalGamma]\ \[Rho]\ \(h\&^\)\ U\&_\^2 + 2\ k\ \[CapitalGamma]\ \[Rho]\ H\&_\ \(u\&^\)\ U\&_ + \[ImaginaryI]\ dpdx\ \(h\&^\) - g\ k\ \[Rho]\ H\&_\ \(h\&^\)\ \(sin(\[Theta])\) - \[ImaginaryI]\ \(\[Tau]\&^\)\_b - \[ImaginaryI]\ \(\[Tau]\&^\)\_w\)\/\(k\ H\&_\)\)\)}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(pressurefluc = p\&^ /. pans[\([1]\)]\)], "Input"], Cell[BoxData[ \(TraditionalForm \`\(-\(\(\(-k\)\ \[CapitalGamma]\ \[Rho]\ \(h\&^\)\ U\&_\^2 + 2\ k\ \[CapitalGamma]\ \[Rho]\ H\&_\ \(u\&^\)\ U\&_ + \[ImaginaryI]\ dpdx\ \(h\&^\) - g\ k\ \[Rho]\ H\&_\ \(h\&^\)\ \(sin(\[Theta])\) - \[ImaginaryI]\ \(\[Tau]\&^\)\_b - \[ImaginaryI]\ \(\[Tau]\&^\)\_w\)\/\(k\ H\&_\)\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["pf1=Expand[pressurefluc]", "Input", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm \`\(\[CapitalGamma]\ \[Rho]\ \(h\&^\)\ U\&_\^2\)\/H\&_ - 2\ \[CapitalGamma]\ \[Rho]\ \(u\&^\)\ U\&_ - \(\[ImaginaryI]\ dpdx\ \(h\&^\)\)\/\(k\ H\&_\) + g\ \[Rho]\ \(h\&^\)\ \(sin(\[Theta])\) + \(\[ImaginaryI]\ \(\[Tau]\&^\)\_b\)\/\(k\ H\&_\) + \(\[ImaginaryI]\ \(\[Tau]\&^\)\_w\)\/\(k\ H\&_\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["pf2=Collect[pf1,k]", "Input", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm \`\(\[CapitalGamma]\ \[Rho]\ \(h\&^\)\ U\&_\^2\)\/H\&_ - 2\ \[CapitalGamma]\ \[Rho]\ \(u\&^\)\ U\&_ + g\ \[Rho]\ \(h\&^\)\ \(sin(\[Theta])\) + \(\(-\(\(\[ImaginaryI]\ dpdx\ \(h\&^\)\)\/H\&_\)\) + \(\[ImaginaryI]\ \(\[Tau]\&^\)\_b\)\/H\&_ + \(\[ImaginaryI]\ \(\[Tau]\&^\)\_w\)\/H\&_\)\/k\)], "Output"] }, Open ]], Cell[TextData[{ "We now substitute the result from the mass balance to eliminate ", Cell[BoxData[ \(TraditionalForm\`u\&^\)]], " into the equation for the pressure." }], "Text"], Cell[CellGroupData[{ Cell["pf3=pf2/.massans[[1]]", "Input", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm \`\(-\(\(\[CapitalGamma]\ \[Rho]\ \(h\&^\)\ U\&_\^2\)\/H\&_\)\) + g\ \[Rho]\ \(h\&^\)\ \(sin(\[Theta])\) + \(\(-\(\(\[ImaginaryI]\ dpdx\ \(h\&^\)\)\/H\&_\)\) + \(\[ImaginaryI]\ \(\[Tau]\&^\)\_b\)\/H\&_ + \(\[ImaginaryI]\ \(\[Tau]\&^\)\_w\)\/H\&_\)\/k\)], "Output"] }, Open ]], Cell["Now make this nondimensional", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(pf4 = Expand[pf3\/\(\[Rho]\ U\&_\^2\)]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm \`\(g\ \(sin(\[Theta])\)\ \(h\&^\)\)\/U\&_\^2 - \(\[CapitalGamma]\ \(h\&^\)\)\/H\&_ - \(\[ImaginaryI]\ dpdx\ \(h\&^\)\)\/\(k\ \[Rho]\ H\&_\ U\&_\^2\) + \(\[ImaginaryI]\ \(\[Tau]\&^\)\_b\)\/\(k\ \[Rho]\ H\&_\ U\&_\^2\) + \(\[ImaginaryI]\ \(\[Tau]\&^\)\_w\)\/\(k\ \[Rho]\ H\&_\ U\&_\^2\)\)], "Output"] }, Open ]], Cell[TextData[{ "We need to translate wavenumber, ", StyleBox["k", FontSlant->"Italic"], ", into the dimensionless wavenumber \[Alpha]." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(pf5 = pf4 /. k \[Rule] \[Alpha]\/H\&_\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm \`\(g\ \(sin(\[Theta])\)\ \(h\&^\)\)\/U\&_\^2 - \(\[CapitalGamma]\ \(h\&^\)\)\/H\&_ - \(\[ImaginaryI]\ dpdx\ \(h\&^\)\)\/\(\[Alpha]\ \[Rho]\ U\&_\^2\) + \(\[ImaginaryI]\ \(\[Tau]\&^\)\_b\)\/\(\[Alpha]\ \[Rho]\ U\&_\^2\) + \(\[ImaginaryI]\ \(\[Tau]\&^\)\_w\)\/\(\[Alpha]\ \[Rho]\ U\&_\^2\)\)], "Output"] }, Open ]], Cell["\<\ We need to eliminate the stress perturbations in terms of the \ average pressure drop. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(pf6 = pf5 /. {\[Tau]\&^\_b \[Rule] 2\ h\&^\ \[Tau]\&_\ /\ H\&_, \[Tau]\&^\_w \[Rule] 2\ h\&^\ \[Tau]\&_\ /\ H\&_}\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm \`\(4\ \[ImaginaryI]\ \[Tau]\&_\ \(h\&^\)\)\/\(\[Alpha]\ \[Rho]\ H\&_\ U\&_\^2\) + \(g\ \(sin(\[Theta])\)\ \(h\&^\)\)\/U\&_\^2 - \(\[CapitalGamma]\ \(h\&^\)\)\/H\&_ - \(\[ImaginaryI]\ dpdx\ \(h\&^\)\)\/\(\[Alpha]\ \[Rho]\ U\&_\^2\)\)], "Output"] }, Open ]], Cell["Here we use the basestate solution again.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(pf7 = Expand[\((pf6\ *H\&_/h\&^\ )\) /. \[Tau]\&_\ \[Rule] \(-\(\(dpdx\ H\&_\)\/2\)\)]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm \`\(-\[CapitalGamma]\) + \(g\ H\&_\ \(sin(\[Theta])\)\)\/U\&_\^2 - \(3\ \[ImaginaryI]\ dpdx\ H\&_\)\/\(\[Alpha]\ \[Rho]\ U\&_\^2\)\)], "Output"] }, Open ]], Cell["We can recover the Froude number, F.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(pf8 = pf7 /. g -> \ F\ U\&_\^2/H\&_\)], "Input"], Cell[BoxData[ \(TraditionalForm \`\(-\[CapitalGamma]\) + F\ \(sin(\[Theta])\) - \(3\ \[ImaginaryI]\ dpdx\ H\&_\)\/\(\[Alpha]\ \[Rho]\ U\&_\^2\)\)], "Output"] }, Open ]], Cell["\<\ We can eliminate the pressure drop in terms of the Reynolds number. \ First we substitute for the pressure drop. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(pf9 = pf8 /. dpdx \[Rule] \(-\(\(U\&_\ 12\ \[Mu]\)\/H\&_\^2\)\)\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm \`\(-\[CapitalGamma]\) + F\ \(sin(\[Theta])\) + \(36\ \[ImaginaryI]\ \[Mu]\)\/\(\[Alpha]\ \[Rho]\ H\&_\ U\&_\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(pf10 = pf9 /. U\&_ \[Rule] \(R\ \[Mu]\)\/\(\[Rho]\ H\&_\)\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm \`\(-\[CapitalGamma]\) + F\ \(sin(\[Theta])\) + \(36\ \[ImaginaryI]\)\/\(R\ \[Alpha]\)\)], "Output"] }, Open ]], Cell[TextData["We would like to know the value of \[CapitalGamma]."], "Text"], Cell[CellGroupData[{ Cell["gamma=Integrate[(6*y - 6*y^2)^2,{y,0,1}]", "Input", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm\`6\/5\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "The final answer for the pressure variation is", "Subsubsection"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(pf11 = pf10 /. \[CapitalGamma] -> gamma\)], "Input"], Cell[BoxData[ \(TraditionalForm \`F\ \(sin(\[Theta])\) + \(36\ \[ImaginaryI]\)\/\(R\ \[Alpha]\) - 6\/5\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "The final answer for the stress variation is", "Subsubsection"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(stressintlam = \(-\((\((\[Tau]\&^ /. tauans[\([1]\)])\) H\&_/h\&^)\)\) /. U\&_ \[Rule] \(R\ \[Mu]\)\/\(\[Rho]\ H\&_\)\)], "Input"], Cell[BoxData[ \(TraditionalForm\`12\/R\)], "Output"] }, Open ]], Cell["\<\ In terms of the notation for the differential problem we have\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(pressureintlam = pf10 /. \[CapitalGamma] -> gamma\)], "Input"], Cell[BoxData[ \(TraditionalForm \`F\ \(sin(\[Theta])\) + \(36\ \[ImaginaryI]\)\/\(R\ \[Alpha]\) - 6\/5\)], "Output"] }, Open ]] }, Open ]] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Pressure and shear stress variations from the linearized ", StyleBox["differential", FontSlant->"Italic"], " equation for a ", StyleBox["polynomial", FontVariations->{"Underline"->True}], " profile (i.e. turbulent) flow. " }], "Section"], Cell[CellGroupData[{ Cell[TextData[{ "Base state ", StyleBox["turbulent", FontSlant->"Italic"], " channel flow" }], "Subsection"], Cell[TextData[{ "We choose a polynomial profile. Brodkey and Hershey,", StyleBox["Transport Phenomena: A Unified Approach", FontSlant->"Italic"], ", McGraw-Hill, 1988, p244, for a consistent average profile that fits the \ boundary conditions. (Note that the much touted 1/7 power profile does not \ make sense nor fit the boundary conditions.) We do not have any Reynolds \ stresses however." }], "Text"], Cell[CellGroupData[{ Cell["\<\ The form of the turbulent profile translated for a domain of 0 to 1 \ (from -1 to 1) is (y-> 2 (y-1/2))\ \>", "Text"], Cell[CellGroupData[{ Cell["tprof1=1+a1 4 (y-1/2)^2 + a2 2^(2 m) (y-1/2)^(2 m)", "Input", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm \`2\^\(2\ m\)\ a2\ \((y - 1\/2)\)\^\(2\ m\) + 4\ a1\ \((y - 1\/2)\)\^2 + 1 \)], "Output"] }, Open ]], Cell["Here are the definitions of a1 and a2", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(tprof2 = tprof1 /. {a1 -> \((s - m)\)/\((m - 1)\), a2 -> \((1 - s)\)/\((m - 1)\)} \)], "Input"], Cell[BoxData[ \(TraditionalForm \`\(2\^\(2\ m\)\ \((1 - s)\)\ \((y - 1\/2)\)\^\(2\ m\)\)\/\(m - 1\) + \(4\ \((s - m)\)\ \((y - 1\/2)\)\^2\)\/\(m - 1\) + 1\)], "Output"] }, Open ]], Cell["We now normalize this so that the average velocity is 1.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(tprof3 = tprof2/\(( Integrate[tprof2, {y, 0, 1}, Assumptions -> {\ \ Re[m] > 0}]/ Integrate[1, {y, 0, 1}])\)\)], "Input"], Cell[BoxData[ \(TraditionalForm \`\(-\(\(6\ \((m - 1)\)\ \((2\ m + 1)\)\ \((\(2\^\(2\ m\)\ \((1 - s)\)\ \((y - 1\/2)\)\^\(2\ m\)\)\/\(m - 1 \) + \(4\ \((s - m)\)\ \((y - 1\/2)\)\^2\)\/\(m - 1\) + 1) \)\)\/\(\(-8\)\ m\^2 - 4\ \((s - 2)\)\ m - 3\ \((\(-1\))\)\^\(2\ m\) + 3\ \((\(-1\))\)\^\(2\ m\)\ s + s + 3 \)\)\)\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Here are the expressions for s and m", "Text"], Cell[BoxData[ \(mm = \ Round[\(- .617\) + .008211\ R^ .786]\ ; \n ss = \ .585\ + .003172\ R^ .833; \)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(tproftest = \((tprof3\ /. {s -> ss, m -> mm})\) /. 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We will need to expand this aroung y=0, (domain \ perturbation) because the wall is not at 0, it is a y= \[Eta] and we don't \ want to have to evaluate the boundary condition there all of the time. \n\n\ This should be\n\n\[Tau](0) = 1/", "R", " (", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\^2\)]], " \[Phi]+\[PartialD]\[Phi]/\[PartialD]y + \[PartialD]", Cell[BoxData[ \(TraditionalForm\`U\&_\)]], "/\[PartialD]y) @ y=0 " }], "Text"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Here we solve the 0 order problem", "Subsection"], Cell[CellGroupData[{ Cell["Here is the first perturbation term:", "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[ \(eqturb0\)], "Input"], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{"\[ImaginaryI]", " ", RowBox[{ SubsuperscriptBox["\[Phi]", "0", TagBox[\((4)\), Derivative], MultilineFunction->None], "(", "y", ")"}]}], "R"], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(eq0turb1 = DSolve[eqturb0 == 0, \[Phi]\_0[y], y]\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{"{", RowBox[{\(\(\[Phi]\_0\)(y)\), "\[Rule]", RowBox[{ RowBox[{ SubscriptBox[ TagBox["c", C], "4"], " ", \(y\^3\)}], "+", RowBox[{ SubscriptBox[ TagBox["c", C], "3"], " ", \(y\^2\)}], "+", RowBox[{ SubscriptBox[ TagBox["c", C], "2"], " ", "y"}], "+", SubscriptBox[ TagBox["c", C], "1"]}]}], "}"}], "}"}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(eq0turb2 = \[Phi]\_0[y] /. eq0turb1\[LeftDoubleBracket]1\[RightDoubleBracket]\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox[ TagBox["c", C], "4"], " ", \(y\^3\)}], "+", RowBox[{ SubscriptBox[ TagBox["c", C], "3"], " ", \(y\^2\)}], "+", RowBox[{ SubscriptBox[ TagBox["c", C], "2"], " ", "y"}], "+", SubscriptBox[ TagBox["c", C], "1"]}], TraditionalForm]], "Output"] }, Open ]], Cell["\<\ The first boundary condition is that the top wall normal velocity \ is 0\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(bcturb01 = eq0turb2 /. {y \[Rule] 1}\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox[ TagBox["c", C], "1"], "+", SubscriptBox[ TagBox["c", C], "2"], "+", SubscriptBox[ TagBox["c", C], "3"], "+", SubscriptBox[ TagBox["c", C], "4"]}], TraditionalForm]], "Output"] }, Open ]], Cell["\<\ The next boundary condition is that the top wall tangential \ velocity is 0\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(bcturb02 = \[PartialD]\_y eq0turb2 /. y \[Rule] 1\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox[ TagBox["c", C], "2"], "+", RowBox[{"2", " ", SubscriptBox[ TagBox["c", C], "3"]}], "+", RowBox[{"3", " ", SubscriptBox[ TagBox["c", C], "4"]}]}], TraditionalForm]], "Output"] }, Open ]], Cell["At the wavy wall, the normal velocity is 0", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(bcturb03 = eq0turb2 /. y \[Rule] 0\)], "Input"], Cell[BoxData[ FormBox[ SubscriptBox[ TagBox["c", C], "1"], TraditionalForm]], "Output"] }, Open ]], Cell[TextData[ "At the wavy wall, the tangential velocity is 0. Note that the wall is not \ always at 0 so we need to use domain perturbation to fix this. It involves \ expanding the tangential velocity around 0.\n\nThe result is that \ u(\[Eta])=0= u'(0) + \[PartialD]", Cell[BoxData[ \(TraditionalForm\`\(\(U\&_\)(0)\)/\[PartialD]y\)]]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(bcturb04a = \((\[PartialD]\_y eq0turb2 + \[PartialD]\_{y, 1}tprof3)\)\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"3", " ", SubscriptBox[ TagBox["c", C], "4"], " ", \(y\^2\)}], "+", RowBox[{"2", " ", SubscriptBox[ TagBox["c", C], "3"], " ", "y"}], "-", \(\(6\ \((m - 1)\)\ \((2\ m + 1)\)\ \((\(2\^\(2\ m + 1\)\ m\ \((1 - s)\)\ \((y - 1\/2)\)\^\(2\ m - 1\)\)\/\(m - 1\) + \(8\ \((s - m)\)\ \((y - 1\/2)\)\)\/\(m - 1\)) \)\)\/\(\(-8\)\ m\^2 - 4\ \((s - 2)\)\ m - 3\ \((\(-1\))\)\^\(2\ m\) + 3\ \((\(-1\))\)\^\(2\ m\)\ s + s + 3 \)\), "+", SubscriptBox[ TagBox["c", C], "2"]}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(bcturb04 = bcturb04a /. y -> 0\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox[ TagBox["c", C], "2"], "-", \(\(6\ \((m - 1)\)\ \((2\ m + 1)\)\ \((\(4\ \((\(-1\))\)\^\(2\ m - 1\)\ m\ \((1 - s)\)\)\/\(m - 1 \) - \(4\ \((s - m)\)\)\/\(m - 1\))\)\)\/\(\(-8\)\ m\^2 - 4\ \((s - 2)\)\ m - 3\ \((\(-1\))\)\^\(2\ m\) + 3\ \((\(-1\))\)\^\(2\ m\)\ s + s + 3\)\)}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(conststurb0 = Solve[{bcturb01 == 0, bcturb02 == 0, bcturb03 == 0, bcturb04 == 0}, { C[1], C[2], C[3], C[4]}]\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{ SubscriptBox[ TagBox["c", C], "1"], "\[Rule]", "0"}], ",", RowBox[{ SubscriptBox[ TagBox["c", C], "2"], "\[Rule]", \(-\(\(24\ \((2\ m + 1)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\)\)\/\(8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3\)\)\)}], ",", RowBox[{ SubscriptBox[ TagBox["c", C], "3"], "\[Rule]", \(\(48\ \((2\ m + 1)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\)\)\/\(8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3\)\)}], ",", RowBox[{ SubscriptBox[ TagBox["c", C], "4"], "\[Rule]", \(-\(\(24\ \((2\ m + 1)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\)\)\/\(8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3\)\)\)}]}], "}"}], "}"}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(phiturb0 = eq0turb2 /. conststurb0[\([1]\)]\)], "Input"], Cell[BoxData[ \(TraditionalForm \`\(-\(\(24\ \((2\ m + 1)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\)\ y\^3\)\/\(8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3 \)\)\) + \(48\ \((2\ m + 1)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\)\ y\^2\)\/\(8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3 \) - \(24\ \((2\ m + 1)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\)\ y\)\/\(8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3 \)\)], "Output"] }, Open ]], Cell[BoxData[ \(\(phi0turbsub = phiturb0 /. {m -> mm, s -> ss}; \)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Short[phi0turbsub]\)], "Input"], Cell[BoxData[ FormBox[ TagBox[ \(\(-\(\(24\ \((2\ \(Round(\[LeftSkeleton]1\[RightSkeleton])\) + 1)\) \[LeftSkeleton]1\[RightSkeleton] \((\[LeftSkeleton]1\[RightSkeleton])\)\ y\^3\)\/\(8\ \(Round(\[LeftSkeleton]1\[RightSkeleton])\)\^2 + \[LeftSkeleton]10\[RightSkeleton]\)\)\) + \[LeftSkeleton]1\[RightSkeleton] - \[LeftSkeleton]1\[RightSkeleton]\/\[LeftSkeleton]1 \[RightSkeleton]\), Short], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Calculate the 0 order pressure", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"pansturb01", "=", RowBox[{"FullSimplify", "[", RowBox[{"pturbeq0", "/.", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox[\(\[Phi]\_0\), "\[Prime]", MultilineFunction->None], "[", "y", "]"}], "\[Rule]", \(\[PartialD]\_y phiturb0\)}], ",", \(\[Phi]\_0[y] \[Rule] phiturb0\), ",", RowBox[{ RowBox[{ SuperscriptBox[\(\[Phi]\_0\), TagBox[\((3)\), Derivative], MultilineFunction->None], "[", "y", "]"}], "\[Rule]", \(\[PartialD]\_{y, 3}phiturb0\)}]}], "}"}]}], "]"}]}]], "Input"], Cell[BoxData[ \(TraditionalForm \`\(-\(\(144\ \[ImaginaryI]\ \((2\ m + 1)\)\ \((\((\(-1\))\)\^\(2\ m\)\ \((s - 1)\)\ m + m - s)\)\)\/\(R\ \((3\ \((\(-1\))\)\^\(2\ m\)\ \((s - 1)\) + s - 4\ m\ \((2\ m + s - 2)\) + 3)\)\)\)\)\)], "Output"] }, Open ]], Cell[TextData[{ "We recall that the first term has ", Cell[BoxData[ \(TraditionalForm\`\(\ \ \[Alpha]\^\(-1\)\)\)]], "power so we write" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(pressureturb0 = pansturb01/\[Alpha]\)], "Input"], Cell[BoxData[ \(TraditionalForm \`\(-\(\(144\ \[ImaginaryI]\ \((2\ m + 1)\)\ \((\((\(-1\))\)\^\(2\ m\)\ 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FractionBox[ RowBox[{"\[ImaginaryI]", " ", RowBox[{ SubsuperscriptBox["\[Phi]", "1", TagBox[\((4)\), Derivative], MultilineFunction->None], "(", "y", ")"}]}], "R"]}], TraditionalForm]], "Output"] }, Open ]], Cell["This next one takes a long time.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(eq1t1 = Simplify[eqturb1 /. {\[Phi]\_0[y] -> phiturb0, \(\(\[Phi]\_0'\)'\)[y] -> D[phiturb0, {y, 2}], \n\t\t\t \(\(u'\)'\)[y] :> \ D[tprof3, {y, 2}], \ u[y] :> \ tprof3}]\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{ \(\((288\ \(( \(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\)\ \((3\ y - 2)\)\ \((\(-\((2\ y - 1)\)\^\(2\ m\)\) + 4\ m\ \((y - 1)\)\ y + s\ \((\((2\ y - 1)\)\^\(2\ m\) - 4\ y\^2 + 4\ y - 1)\) + 1) \)\ \((2\ m + 1)\)\^2)\)/ \((\(-8\)\ m\^2 - 4\ \((s - 2)\)\ m - 3\ \((\(-1\))\)\^\(2\ m\) + 3\ \((\(-1\))\)\^\(2\ m\)\ s + s + 3)\)\^2\), "-", \(\((1152\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\)\ \((y - 1)\)\^2\ y\ \((2\ m\^2\ \((s 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m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3)\)\^2) \)\)\) + \((288\ \[ImaginaryI]\ \((2\ m + 1)\)\^2\ \((6\ m\^2 - m - 10)\)\ R\ \((s - 1)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\)\ y\^4) \)/\((\((m + 1)\)\ \((2\ m + 3)\)\ \((2\ m + 5)\)\ \((4\ m + 8)\)\ \((8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3)\)\^2) \) - \(( 36\ \[ImaginaryI]\ \((104\ m\^4 + 84\ m\^3 - 202\ m\^2 - 189\ m - 40) \)\ R\ \((s - 1)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\)\ y\^3) \)/\((\((m + 1)\)\ \((m + 2)\)\ \((2\ m + 5)\)\ \((4\ m + 6)\)\ \((8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3)\)\^2) \) + \(( 432\ \[ImaginaryI]\ \((8\ m\^4 - 34\ m\^2 - 26\ m - 5)\)\ R\ \((s - 1)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\)\ y\^2) \)/\((\((m + 2)\)\ \((2\ m + 3)\)\ \((2\ m + 5)\)\ \((4\ m + 4)\)\ \((8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3)\)\^2) \) - \(( 9\ \[ImaginaryI]\ \((2\ m + 1)\)\^2\ \((16\ m\^3 - 64\ m\^2 - 190\ m - 65)\)\ R\ \((s - 1)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\)\ y)\)/ \((\((m + 1)\)\ \((m + 2)\)\ \((2\ m + 3)\)\ \((2\ m + 5)\)\ \((4\ m + 2)\)\ \((8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3)\)\^2) \) - \(( 9\ \[ImaginaryI]\ \((2\ m + 1)\)\^2\ \((4\ m + 7)\)\ R \ \((s - 1)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\))\)/ \((\((m + 1)\)\ \((m + 2)\)\ \((2\ m + 3)\)\ \((2\ m + 5)\)\ \((8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3)\)\^2)\)) \)\ \((2\ y - 1)\)\^\(2\ m\)\), "+", \(\(96\ \[ImaginaryI]\ \((2\ m + 1)\)\^2\ R\ \((m - s)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\)\ y\^7\)\/\(35 \ \((8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3)\)\^2\)\), "-", \(\(48\ \[ImaginaryI]\ \((2\ m + 1)\)\^2\ R\ \((m - s)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\)\ y\^6\)\/\(5\ \((8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3)\)\^2\)\), "+", \(\(12\ \[ImaginaryI]\ \((2\ m + 1)\)\^2\ R\ \((4\ m - 7\ s + 3)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\)\ y\^5\)\/\(5\ \((8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3)\)\^2\)\), "+", \(\(24\ \[ImaginaryI]\ \((2\ m + 1)\)\^2\ R\ \((s - 1)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\)\ y\^4\)\/\(( 8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3)\)\^2\), "+", SubscriptBox[ TagBox["c", C], "1"], "+", RowBox[{"y", " ", SubscriptBox[ TagBox["c", C], "2"]}], "+", RowBox[{\(y\^2\), " ", SubscriptBox[ TagBox["c", C], "3"]}], "+", RowBox[{\(y\^3\), " ", SubscriptBox[ TagBox["c", C], "4"]}]}]}], "}"}], "}"}], TraditionalForm]], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(eq1t3 = \[Phi]\_1[y] /. eq1t2\[LeftDoubleBracket]1\[RightDoubleBracket]\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{ \(\((\(-\(\((144\ \[ImaginaryI]\ \((2\ m - 3)\)\ \((2\ m + 1)\)\^2\ R\ \((s - 1)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\)\ y\^5)\)/ \((\((m + 2)\)\ \((2\ m + 3)\)\ \((4\ m + 10)\)\ \((8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3)\)\^2) \)\)\) + \((288\ \[ImaginaryI]\ \((2\ m + 1)\)\^2\ \((6\ m\^2 - m - 10)\)\ R\ \((s - 1)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\)\ y\^4)\)/ \((\((m + 1)\)\ \((2\ m + 3)\)\ \((2\ m + 5)\)\ \((4\ m + 8)\)\ \((8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3)\)\^2)\) - \((36\ \[ImaginaryI]\ \((104\ m\^4 + 84\ m\^3 - 202\ m\^2 - 189\ m - 40)\)\ R\ \((s - 1)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\)\ y\^3)\)/ \((\((m + 1)\)\ \((m + 2)\)\ \((2\ m + 5)\)\ \((4\ m + 6)\)\ \((8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3)\)\^2)\) + \((432\ \[ImaginaryI]\ \((8\ m\^4 - 34\ m\^2 - 26\ m - 5)\)\ R\ \((s - 1)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\)\ y\^2)\)/ \((\((m + 2)\)\ \((2\ m + 3)\)\ \((2\ m + 5)\)\ \((4\ m + 4)\)\ \((8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3)\)\^2)\) - \((9\ \[ImaginaryI]\ \((2\ m + 1)\)\^2\ \((16\ m\^3 - 64\ m\^2 - 190\ m - 65)\)\ R\ \((s - 1)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\)\ y)\)/ \((\((m + 1)\)\ \((m + 2)\)\ \((2\ m + 3)\)\ \((2\ m + 5)\)\ \((4\ m + 2)\)\ \((8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3)\)\^2)\) - \((9\ \[ImaginaryI]\ \((2\ m + 1)\)\^2\ \((4\ m + 7)\)\ R\ \((s - 1)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\))\)/ \((\((m + 1)\)\ \((m + 2)\)\ \((2\ m + 3)\)\ \((2\ m + 5)\)\ \((8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3)\)\^2)\))\)\ \((2\ y - 1)\)\^\(2\ m\)\), "+", \(\(96\ \[ImaginaryI]\ \((2\ m + 1)\)\^2\ R\ \((m - s)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\)\ y\^7\)\/\(35\ \((8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3)\)\^2\)\), "-", \(\(48\ \[ImaginaryI]\ \((2\ m + 1)\)\^2\ R\ \((m - s)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\)\ y\^6\)\/\(5\ \((8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3)\)\^2\)\), "+", \(\(12\ \[ImaginaryI]\ \((2\ m + 1)\)\^2\ R\ \((4\ m - 7\ s + 3)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\)\ y\^5\)\/\(5\ \((8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3)\)\^2\)\), "+", \(\(24\ \[ImaginaryI]\ \((2\ m + 1)\)\^2\ R\ \((s - 1)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\)\ y\^4\)\/\(( 8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3)\)\^2\), "+", SubscriptBox[ TagBox["c", C], "1"], "+", RowBox[{"y", " ", SubscriptBox[ TagBox["c", C], "2"]}], "+", RowBox[{\(y\^2\), " ", SubscriptBox[ TagBox["c", C], "3"]}], "+", RowBox[{\(y\^3\), " ", SubscriptBox[ TagBox["c", C], "4"]}]}], TraditionalForm]], "Output"] }, Open ]], Cell["\<\ The first boundary condition is that the top wall normal velocity \ is 0\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(bcturb11 = eq1t3 /. {y \[Rule] 1}\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{ \(\(12\ \[ImaginaryI]\ R\ \((4\ m - 7\ s + 3)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\)\ \((2\ m + 1)\)\^2\)\/\(5\ \((8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3)\)\^2\)\), "-", \(\(48\ \[ImaginaryI]\ R\ \((m - s)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\)\ \((2\ m + 1)\)\^2\)\/\(7\ \((8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3)\)\^2\)\), "-", \(\((9\ \[ImaginaryI]\ \((4\ m + 7)\)\ R\ \((s - 1)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\)\ \((2\ m + 1)\)\^2) \)/\((\((m + 1)\)\ \((m + 2)\)\ \((2\ m + 3)\)\ \((2\ m + 5)\)\ \((8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3)\)\^2)\)\), "+", \(\((288\ \[ImaginaryI]\ \((6\ m\^2 - m - 10)\)\ R\ \((s - 1)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\)\ \((2\ m + 1)\)\^2) \)/\((\((m + 1)\)\ \((2\ m + 3)\)\ \((2\ m + 5)\)\ \((4\ m + 8)\)\ \((8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3)\)\^2)\)\), "-", \(\((9\ \[ImaginaryI]\ \((16\ m\^3 - 64\ m\^2 - 190\ m - 65)\)\ R\ \((s - 1)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\)\ \((2\ m + 1)\)\^2) \)/\((\((m + 1)\)\ \((m + 2)\)\ \((2\ m + 3)\)\ \((2\ m + 5)\)\ \((4\ m + 2)\)\ \((8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3)\)\^2)\)\), "-", \(\((144\ \[ImaginaryI]\ \((2\ m - 3)\)\ R\ \((s - 1)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\)\ \((2\ m + 1)\)\^2) \)/\((\((m + 2)\)\ \((2\ m + 3)\)\ \((4\ m + 10)\)\ \((8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3)\)\^2)\)\), "+", \(\(24\ \[ImaginaryI]\ R\ \((s - 1)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\)\ \((2\ m + 1)\)\^2\)\/\(( 8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3)\)\^2\), "+", \(\((432\ \[ImaginaryI]\ \((8\ m\^4 - 34\ m\^2 - 26\ m - 5)\)\ R\ \((s - 1)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\))\)/ \((\((m + 2)\)\ \((2\ m + 3)\)\ \((2\ m + 5)\)\ \((4\ m + 4)\)\ \((8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3)\)\^2)\)\), "-", \(\((36\ \[ImaginaryI]\ \((104\ m\^4 + 84\ m\^3 - 202\ m\^2 - 189\ m - 40)\)\ R\ \((s - 1)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\))\)/ \((\((m + 1)\)\ \((m + 2)\)\ \((2\ m + 5)\)\ \((4\ m + 6)\)\ \((8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3)\)\^2)\)\), "+", SubscriptBox[ TagBox["c", C], "1"], "+", SubscriptBox[ TagBox["c", C], "2"], "+", SubscriptBox[ TagBox["c", C], "3"], "+", SubscriptBox[ TagBox["c", C], "4"]}], TraditionalForm]], "Output"] }, Open ]], Cell["\<\ The next boundary condition is that the top wall tangential \ velocity is 0\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(bcturb12 = \[PartialD]\_y eq1t3 /. y \[Rule] 1; \)\)], "Input"], Cell[BoxData[ \(Short[bcturb12]\)], "Input"], Cell[BoxData[ FormBox[ TagBox[ RowBox[{\(\[LeftSkeleton]15\[RightSkeleton]\), "+", RowBox[{"3", " ", SubscriptBox[ TagBox["c", C], "4"]}]}], Short], TraditionalForm]], "Output"] }, Open ]], Cell["At the wavy wall, the normal velocity is 0", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(bcturb13 = eq1t3 /. y \[Rule] 0\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox[ TagBox["c", C], "1"], "-", \(\((9\ \[ImaginaryI]\ \((\(-1\))\)\^\(2\ m\)\ \((2\ m + 1)\)\^2\ \((4\ m + 7)\)\ R\ \((s - 1)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\))\)/ \((\((m + 1)\)\ \((m + 2)\)\ \((2\ m + 3)\)\ \((2\ m + 5)\)\ \((8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3)\)\^2)\)\)}], TraditionalForm]], "Output"] }, Open ]], Cell[TextData[{ "At the wavy wall, the tangential velocity is 0.\nAt first order we have \ that \[PartialD]", Cell[BoxData[ \(TraditionalForm\`\[Phi]\_1\)]], "/\[PartialD]y = 0. The average velocity is only in the lowest order \ \[Alpha] equation. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(bcturb14 = \((\[PartialD]\_y eq1t3)\) /. y -> 0\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{ \(-\(\((36\ \[ImaginaryI]\ \((\(-1\))\)\^\(2\ m - 1\)\ m\ \((4\ m + 7)\)\ R\ \((s - 1)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\)\ \((2\ m + 1)\)\^2)\)/ \((\((m + 1)\)\ \((m + 2)\)\ \((2\ m + 3)\)\ \((2\ m + 5)\)\ \((8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3)\)\^2)\)\)\), "-", \(\((9\ \[ImaginaryI]\ \((\(-1\))\)\^\(2\ m\)\ \((16\ m\^3 - 64\ m\^2 - 190\ m - 65)\)\ R\ \((s - 1)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\)\ \((2\ m + 1)\)\^2) \)/\((\((m + 1)\)\ \((m + 2)\)\ \((2\ m + 3)\)\ \((2\ m + 5)\)\ \((4\ m + 2)\)\ \((8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3)\)\^2)\)\), "+", SubscriptBox[ TagBox["c", C], "2"]}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(conststurb1 = Simplify[Solve[{bcturb11 == 0, bcturb12 == 0, bcturb13 == 0, bcturb14 == 0}, {C[1], C[2], C[3], C[4]}]]; 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We include a perturbation parameter , \[Epsilon], which \ is giving an idea how large the waves are. We need \[Epsilon] to be small \ compared to ", StyleBox["H", FontSlant->"Italic"], ", the channel height, to make the linearization assumption be valid. 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\[Epsilon]\ \(h(x)\))\), " ", RowBox[{"(", RowBox[{"dpdx", "-", RowBox[{ "g", " ", "\[Epsilon]", " ", "\[Rho]", " ", \(sin(\[Theta])\), " ", RowBox[{ SuperscriptBox["h", "\[Prime]", MultilineFunction->None], "(", "x", ")"}]}], "+", RowBox[{"\[Epsilon]", " ", RowBox[{ SuperscriptBox["p", "\[Prime]", MultilineFunction->None], "(", "x", ")"}]}]}], ")"}]}], "\[Rho]"], "+", RowBox[{"\[CapitalGamma]", " ", RowBox[{"(", RowBox[{ RowBox[{ "2", " ", "\[Epsilon]", " ", \((H\&_ - \[Epsilon]\ \(h(x)\))\), " ", \((U\&_ + \[Epsilon]\ \(u(x)\))\), " ", RowBox[{ SuperscriptBox["u", "\[Prime]", MultilineFunction->None], "(", "x", ")"}]}], "-", RowBox[{ "\[Epsilon]", " ", \(\((U\&_ + \[Epsilon]\ \(u(x)\))\)\^2\), " ", RowBox[{ SuperscriptBox["h", "\[Prime]", MultilineFunction->None], "(", "x", ")"}]}]}], ")"}]}]}], TraditionalForm]], "Output"] }, Open ]], Cell[TextData[ "This equation is expanded and the individual powers of \[Epsilon] are \ collected."], "Text"], Cell["ex1=Expand[momeq];", "Input"], Cell[TextData["eqs1=Collect[ex1,\[Epsilon]];"], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["First solve the basestate equation", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData["basestate1=Coefficient[eqs1,\[Epsilon],0]"], "Input"], Cell[BoxData[ \(TraditionalForm \`\(dpdx\ H\&_\)\/\[Rho] + \[Tau]\_b0\/\[Rho] + \[Tau]\_w0\/\[Rho]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(basestate2 = basestate1 /. {\[Tau]\_w0 \[Rule] \[Tau]\_b0}\)], "Input"], Cell[BoxData[ \(TraditionalForm\`\(dpdx\ H\&_\)\/\[Rho] + \(2\ \[Tau]\_b0\)\/\[Rho]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(wallshearstress = Solve[basestate2 == 0, \[Tau]\_b0]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`{{\[Tau]\_b0 \[Rule] \(-\(\(dpdx\ H\&_\)\/2\)\)}}\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Calculation of the shear stress", "Subsubsection"], Cell["\<\ We will need to eliminate the shear fluctuations in terms of \ average shear and the heights. The way that this can be done is to assume \ that the spatial variations in the flow is slow enough that the average \ relation for the friction factor (or wall shear stress) holds at any location \ along the wave. This is implemented by expanding the average relation in \ terms of a average and a perturbation equation. Start with the average relation for shear stress and Reynolds number. \ \>", "Text"], Cell[BoxData[ \(turbtaueq1 = \[Tau] - \ .079\ \((\ \(\(\[Mu]\ /\[Rho]\)/U\)/H)\)\^ .25\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{"\[Tau]", "-", RowBox[{"0.0790000000000000035`", " ", SuperscriptBox[\((\[Mu]\/\(H\ U\ \[Rho]\))\), StyleBox["0.25`", StyleBoxAutoDelete->True, PrintPrecision->2]]}]}], TraditionalForm]], "Output"], Cell["Now expand. ", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(turbtaueq2 = turbtaueq1 /. {\[Tau] -> \[Tau]\&_ + \ \[Epsilon]\ \[Tau]\&^, \ H -> H\&_ + \ \[Epsilon]\ h\&^, \ U -> U\&_\ + \n\t\t\t\t\t\[Epsilon]\ u\&^}\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{\(\[Tau]\&_\), "+", \(\[Epsilon]\ \(\[Tau]\&^\)\), "-", RowBox[{"0.0790000000000000035`", " ", SuperscriptBox[ \((\[Mu]\/\(\[Rho]\ \((H\&_ + \[Epsilon]\ \(h\&^\))\)\ \((U\&_ + \[Epsilon]\ \(u\&^\))\)\))\), StyleBox["0.25`", StyleBoxAutoDelete->True, PrintPrecision->2]]}]}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(turbtaueq3 = Series[turbtaueq2, {\[Epsilon], 0, 2}]\)], "Input"], Cell[BoxData[ FormBox[ InterpretationBox[ RowBox[{ RowBox[{"(", RowBox[{\(\[Tau]\&_\), "-", RowBox[{"0.0790000000000000035`", " ", SuperscriptBox[\((\[Mu]\/\(\[Rho]\ H\&_\ U\&_\))\), StyleBox["0.25`", StyleBoxAutoDelete->True, PrintPrecision->2]]}]}], ")"}], "+", \(\((\(\[Tau]\&^\) - \(0.01975`\ \[Mu]\ \((\(-\(\(h\&^\)\/\(H\&_\^2\ U\&_\)\)\) - \(u\&^\)\/\(H\&_\ U\&_\^2\))\)\)\/\(\[Rho]\ \((\[Mu]\/\(\[Rho]\ H\&_\ U\&_\))\)\^0.75`\))\)\ \[Epsilon]\), "-", RowBox[{"0.0790000000000000035`", " ", RowBox[{"(", RowBox[{ FractionBox[ RowBox[{ StyleBox["0.25`", StyleBoxAutoDelete->True, PrintPrecision->2], " ", "\[Mu]", " ", \((\(h\&^\)\^2\/\(H\&_\^3\ U\&_\) + \(\(u\&^\)\ \(h\&^\)\)\/\(H\&_\^2\ U\&_\^2\) + \(u\&^\)\^2\/\(H\&_\ U\&_\^3\))\)}], \(\[Rho]\ \((\[Mu]\/\(\[Rho]\ H\&_\ U\&_\))\)\^0.75`\)], "-", \(\(0.09375`\ \[Mu]\^2\ \((\(-\(\(h\&^\)\/\(H\&_\^2\ U\&_\)\)\) - \(u\&^\)\/\(H\&_\ U\&_\^2\))\)\^2\)\/\(\[Rho]\^2\ \((\[Mu]\/\(\[Rho]\ H\&_\ U\&_\))\)\^1.75`\)\)}], ")"}], " ", \(\[Epsilon]\^2\)}], "+", InterpretationBox[\(O(\[Epsilon]\^3)\), SeriesData[ \[Epsilon], 0, {}, 0, 3, 1]]}], SeriesData[ \[Epsilon], 0, { Plus[ Times[ -0.079000000000000001, Power[ Times[ \[Mu], Power[ \[Rho], -1], Power[ OverBar[ H], -1], Power[ OverBar[ U], -1]], 0.25]], OverBar[ \[Tau]]], Plus[ Times[ -0.01975, \[Mu], Power[ \[Rho], -1], Power[ Times[ \[Mu], Power[ \[Rho], -1], Power[ OverBar[ H], -1], Power[ OverBar[ U], -1]], -0.75], Plus[ Times[ -1, Power[ OverBar[ H], -2], Power[ OverBar[ U], -1], OverHat[ h]], Times[ -1, Power[ OverBar[ H], -1], Power[ OverBar[ U], -2], OverHat[ u]]]], OverHat[ \[Tau]]], Times[ -0.079000000000000001, Plus[ Times[ -0.09375, Power[ \[Mu], 2], Power[ \[Rho], -2], Power[ Times[ \[Mu], Power[ \[Rho], -1], Power[ OverBar[ H], -1], Power[ OverBar[ U], -1]], -1.75], Power[ Plus[ Times[ -1, Power[ OverBar[ H], -2], Power[ OverBar[ U], -1], OverHat[ h]], Times[ -1, Power[ OverBar[ H], -1], Power[ OverBar[ U], -2], OverHat[ u]]], 2]], Times[ 0.25, \[Mu], Power[ \[Rho], -1], Power[ Times[ \[Mu], Power[ \[Rho], -1], Power[ OverBar[ H], -1], Power[ OverBar[ U], -1]], -0.75], Plus[ Times[ Power[ OverBar[ H], -3], Power[ OverBar[ U], -1], Power[ OverHat[ h], 2]], Times[ Power[ OverBar[ H], -2], Power[ OverBar[ U], -2], OverHat[ h], OverHat[ u]], Times[ Power[ OverBar[ H], -1], Power[ OverBar[ U], -3], Power[ OverHat[ u], 2]]]]]]}, 0, 3, 1]], TraditionalForm]], "Output"] }, Open ]], Cell["We first recover the average equation.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(turbaveragetau = Coefficient[turbtaueq3, \[Epsilon], 0]\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{\(\[Tau]\&_\), "-", RowBox[{"0.0790000000000000035`", " ", SuperscriptBox[\((\[Mu]\/\(\[Rho]\ H\&_\ U\&_\))\), StyleBox["0.25`", StyleBoxAutoDelete->True, PrintPrecision->2]]}]}], TraditionalForm]], "Output"] }, Open ]], Cell["We next obtain the first order term", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(turbtau10 = \ Coefficient[turbtaueq3, \[Epsilon], 1]\)], "Input"], Cell[BoxData[ \(TraditionalForm \`\(\[Tau]\&^\) - \(0.01975`\ \[Mu]\ \((\(-\(\(h\&^\)\/\(H\&_\^2\ U\&_\)\)\) - \(u\&^\)\/\(H\&_\ U\&_\^2\))\)\)\/\(\[Rho]\ \((\[Mu]\/\(\[Rho]\ H\&_\ U\&_\))\)\^0.75`\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(turbtau11 = turbtau10 /. massans[\([1]\)]\)], "Input"], Cell[BoxData[ \(TraditionalForm \`\(0.0395000000000000017`\ \[Mu]\ \(h\&^\)\)\/\(\[Rho]\ H\&_\^2\ \((\[Mu]\/\(\[Rho]\ H\&_\ U\&_\))\)\^0.75`\ U\&_\) + \(\[Tau]\&^\)\)], "Output"] }, Open ]], Cell["\<\ This relation for the perturbation shear stress is the same for \ both the top and bottom walls. Note that the perturbation shear stress is \ twice the average shear stress.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(turbtauans = FullSimplify[Solve[turbtau11 == 0, \ \[Tau]\&^]]\)], "Input"], Cell[BoxData[ \(TraditionalForm \`{{\(\[Tau]\&^\) \[Rule] \(-\(\(0.0395000000000000017`\ \((\[Mu]\/\(\[Rho]\ H\&_\ U\&_\))\)\^0.25`\ \(h\&^\)\)\/H\&_ \)\)}}\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Now solve the linear perturbation problem", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData["eq1=Coefficient[eqs1,\[Epsilon],1]"], "Input"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(-\[CapitalGamma]\), " ", RowBox[{ SuperscriptBox["h", "\[Prime]", MultilineFunction->None], "(", "x", ")"}], " ", \(U\&_\^2\)}], "+", RowBox[{"2", " ", "\[CapitalGamma]", " ", \(H\&_\), " ", RowBox[{ SuperscriptBox["u", "\[Prime]", MultilineFunction->None], "(", "x", ")"}], " ", \(U\&_\)}], "-", \(\(dpdx\ \(h(x)\)\)\/\[Rho]\), "+", \(\(\(\[Tau]\_b\)(x)\)\/\[Rho]\), "+", \(\(\(\[Tau]\_w\)(x)\)\/\[Rho]\), "-", RowBox[{"g", " ", \(H\&_\), " ", \(sin(\[Theta])\), " ", RowBox[{ SuperscriptBox["h", "\[Prime]", MultilineFunction->None], "(", "x", ")"}]}], "+", FractionBox[ RowBox[{\(H\&_\), " ", RowBox[{ SuperscriptBox["p", "\[Prime]", MultilineFunction->None], "(", "x", ")"}]}], "\[Rho]"]}], TraditionalForm]], "Output"] }, Open ]], Cell["\<\ We now use a normal mode expansion, that is, assume that the shape \ of the surface, h[x], can be written as one (or a whole sum of) sinuous \ modes. Of course, since the equations are linear, these modes will be \ independent. We use complex exponentials since they will cancel out easier.\ \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"eq2", "=", RowBox[{"eq1", "/.", RowBox[{"{", RowBox[{ \(h[x] \[Rule] h\&^\ Exp[I\ k\ x]\), ",", \(\[Tau]\_w[x] \[Rule] \[Tau]\&^\_w\ \ Exp[I\ k\ x]\), ",", \(\[Tau]\_b[x] \[Rule] \[Tau]\&^\_b\ \ Exp[I\ k\ x]\), ",", RowBox[{ RowBox[{ SuperscriptBox["h", "\[Prime]", MultilineFunction->None], "[", "x", "]"}], "\[Rule]", \(\[PartialD]\_x\((h\&^\ Exp[I\ k\ x])\)\)}], ",", RowBox[{ RowBox[{ SuperscriptBox["p", "\[Prime]", MultilineFunction->None], "[", "x", "]"}], "\[Rule]", \(\[PartialD]\_x\((p\&^\ Exp[I\ k\ x])\)\)}], ",", RowBox[{ RowBox[{ SuperscriptBox["u", "\[Prime]", MultilineFunction->None], "[", "x", "]"}], "\[Rule]", \(\[PartialD]\_x\((u\&^\ Exp[I\ k\ x])\)\)}]}], "}"}]}]}]], "Input"], Cell[BoxData[ \(TraditionalForm \`\(-\[ImaginaryI]\)\ \[ExponentialE]\^\(\[ImaginaryI]\ k\ x\)\ k\ \[CapitalGamma]\ \(h\&^\)\ U\&_\^2 + 2\ \[ImaginaryI]\ \[ExponentialE]\^\(\[ImaginaryI]\ k\ x\)\ k\ \[CapitalGamma]\ H\&_\ \(u\&^\)\ U\&_ - \(dpdx\ \[ExponentialE]\^\(\[ImaginaryI]\ k\ x\)\ \(h\&^\)\)\/\[Rho] + \(\[ImaginaryI]\ \[ExponentialE]\^\(\[ImaginaryI]\ k\ x\)\ k\ H\&_\ \(p\&^\)\)\/\[Rho] - \[ImaginaryI]\ \[ExponentialE]\^\(\[ImaginaryI]\ k\ x\)\ g\ k\ H\&_\ \(h\&^\)\ \(sin(\[Theta])\) + \(\[ExponentialE]\^\(\[ImaginaryI]\ k\ x\)\ \(\[Tau]\&^\)\_b\)\/\[Rho] + \(\[ExponentialE]\^\(\[ImaginaryI]\ k\ x\)\ \(\[Tau]\&^\)\_w\)\/\[Rho]\)], "Output"] }, Open ]], Cell["\<\ It is worth mentioning at this point, that this is no longer a \ differential equation. The restriction of the solution to a periodic domain \ is enough to remove the first derivatives in x. The higher order \ x-derivatives were removed in the \"averaging\" (i.e., when we decided to \ just do long waves and when we integrated in the normal direction. \ \>", "Text"], Cell[CellGroupData[{ Cell["eq3=Expand[eq2/Exp[I k x]]", "Input"], Cell[BoxData[ \(TraditionalForm \`\(-\[ImaginaryI]\)\ k\ \[CapitalGamma]\ \(h\&^\)\ U\&_\^2 + 2\ \[ImaginaryI]\ k\ \[CapitalGamma]\ H\&_\ \(u\&^\)\ U\&_ - \(dpdx\ \(h\&^\)\)\/\[Rho] + \(\[ImaginaryI]\ k\ H\&_\ \(p\&^\)\)\/\[Rho] - \[ImaginaryI]\ g\ k\ H\&_\ \(h\&^\)\ \(sin(\[Theta])\) + \(\[Tau]\&^\)\_b\/\[Rho] + \(\[Tau]\&^\)\_w\/\[Rho]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(pans = Solve[eq3 == 0, p\&^]\)], "Input"], Cell[BoxData[ \(TraditionalForm \`{{\(p\&^\) \[Rule] \(-\(\(\(-k\)\ \[CapitalGamma]\ \[Rho]\ \(h\&^\)\ U\&_\^2 + 2\ k\ \[CapitalGamma]\ \[Rho]\ H\&_\ \(u\&^\)\ U\&_ + \[ImaginaryI]\ dpdx\ \(h\&^\) - g\ k\ \[Rho]\ H\&_\ \(h\&^\)\ \(sin(\[Theta])\) - \[ImaginaryI]\ \(\[Tau]\&^\)\_b - \[ImaginaryI]\ \(\[Tau]\&^\)\_w\)\/\(k\ H\&_\)\)\)}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(pressurefluc = p\&^ /. pans[\([1]\)]\)], "Input"], Cell[BoxData[ \(TraditionalForm \`\(-\(\(\(-k\)\ \[CapitalGamma]\ \[Rho]\ \(h\&^\)\ U\&_\^2 + 2\ k\ \[CapitalGamma]\ \[Rho]\ H\&_\ \(u\&^\)\ U\&_ + \[ImaginaryI]\ dpdx\ \(h\&^\) - g\ k\ \[Rho]\ H\&_\ \(h\&^\)\ \(sin(\[Theta])\) - \[ImaginaryI]\ \(\[Tau]\&^\)\_b - \[ImaginaryI]\ \(\[Tau]\&^\)\_w\)\/\(k\ H\&_\)\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["pf1=Expand[pressurefluc]", "Input"], Cell[BoxData[ \(TraditionalForm \`\(\[CapitalGamma]\ \[Rho]\ \(h\&^\)\ U\&_\^2\)\/H\&_ - 2\ \[CapitalGamma]\ \[Rho]\ \(u\&^\)\ U\&_ - \(\[ImaginaryI]\ dpdx\ \(h\&^\)\)\/\(k\ H\&_\) + g\ \[Rho]\ \(h\&^\)\ \(sin(\[Theta])\) + \(\[ImaginaryI]\ \(\[Tau]\&^\)\_b\)\/\(k\ H\&_\) + \(\[ImaginaryI]\ \(\[Tau]\&^\)\_w\)\/\(k\ H\&_\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["pf2=Collect[pf1,k]", "Input"], Cell[BoxData[ \(TraditionalForm \`\(\[CapitalGamma]\ \[Rho]\ \(h\&^\)\ U\&_\^2\)\/H\&_ - 2\ \[CapitalGamma]\ \[Rho]\ \(u\&^\)\ U\&_ + g\ \[Rho]\ \(h\&^\)\ \(sin(\[Theta])\) + \(\(-\(\(\[ImaginaryI]\ dpdx\ \(h\&^\)\)\/H\&_\)\) + \(\[ImaginaryI]\ \(\[Tau]\&^\)\_b\)\/H\&_ + \(\[ImaginaryI]\ \(\[Tau]\&^\)\_w\)\/H\&_\)\/k\)], "Output"] }, Open ]], Cell[TextData[{ "We now substitute the result from the mass balance to eliminate ", Cell[BoxData[ \(TraditionalForm\`u\&^\)]], " into the equation for the pressure." }], "Text"], Cell[CellGroupData[{ Cell["pf3=pf2/.massans[[1]]", "Input"], Cell[BoxData[ \(TraditionalForm \`\(-\(\(\[CapitalGamma]\ \[Rho]\ \(h\&^\)\ U\&_\^2\)\/H\&_\)\) + g\ \[Rho]\ \(h\&^\)\ \(sin(\[Theta])\) + \(\(-\(\(\[ImaginaryI]\ dpdx\ \(h\&^\)\)\/H\&_\)\) + \(\[ImaginaryI]\ \(\[Tau]\&^\)\_b\)\/H\&_ + \(\[ImaginaryI]\ \(\[Tau]\&^\)\_w\)\/H\&_\)\/k\)], "Output"] }, Open ]], Cell["Now make this nondimensional", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(pf4 = Expand[pf3\/\(\[Rho]\ U\&_\^2\)]\)], "Input"], Cell[BoxData[ \(TraditionalForm \`\(g\ \(sin(\[Theta])\)\ \(h\&^\)\)\/U\&_\^2 - \(\[CapitalGamma]\ \(h\&^\)\)\/H\&_ - \(\[ImaginaryI]\ dpdx\ \(h\&^\)\)\/\(k\ \[Rho]\ H\&_\ U\&_\^2\) + \(\[ImaginaryI]\ \(\[Tau]\&^\)\_b\)\/\(k\ \[Rho]\ H\&_\ U\&_\^2\) + \(\[ImaginaryI]\ \(\[Tau]\&^\)\_w\)\/\(k\ \[Rho]\ H\&_\ U\&_\^2\)\)], "Output"] }, Open ]], Cell[TextData[{ "We need to translate wavenumber, ", StyleBox["k", FontSlant->"Italic"], ", into the dimensionless wavenumber \[Alpha]." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(pf5 = pf4 /. k \[Rule] \[Alpha]\/H\&_\)], "Input"], Cell[BoxData[ \(TraditionalForm \`\(g\ \(sin(\[Theta])\)\ \(h\&^\)\)\/U\&_\^2 - \(\[CapitalGamma]\ \(h\&^\)\)\/H\&_ - \(\[ImaginaryI]\ dpdx\ \(h\&^\)\)\/\(\[Alpha]\ \[Rho]\ U\&_\^2\) + \(\[ImaginaryI]\ \(\[Tau]\&^\)\_b\)\/\(\[Alpha]\ \[Rho]\ U\&_\^2\) + \(\[ImaginaryI]\ \(\[Tau]\&^\)\_w\)\/\(\[Alpha]\ \[Rho]\ U\&_\^2\)\)], "Output"] }, Open ]], Cell["\<\ We need to eliminate the stress perturbations in terms of the \ average pressure drop. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(pf6 = pf5 /. {\[Tau]\&^\_b \[Rule] 2\ h\&^\ \[Tau]\&_\ /\ H\&_, \[Tau]\&^\_w \[Rule] 2\ h\&^\ \[Tau]\&_\ /\ H\&_}\)], "Input"], Cell[BoxData[ \(TraditionalForm \`\(4\ \[ImaginaryI]\ \[Tau]\&_\ \(h\&^\)\)\/\(\[Alpha]\ \[Rho]\ H\&_\ U\&_\^2\) + \(g\ \(sin(\[Theta])\)\ \(h\&^\)\)\/U\&_\^2 - \(\[CapitalGamma]\ \(h\&^\)\)\/H\&_ - \(\[ImaginaryI]\ dpdx\ \(h\&^\)\)\/\(\[Alpha]\ \[Rho]\ U\&_\^2\)\)], "Output"] }, Open ]], Cell["Here we use the basestate solution again.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(pf7 = Expand[\((pf6\ *H\&_/h\&^\ )\) /. \[Tau]\&_\ \[Rule] \(-\(\(dpdx\ H\&_\)\/2\)\)]\)], "Input"], Cell[BoxData[ \(TraditionalForm \`\(-\[CapitalGamma]\) + \(g\ H\&_\ \(sin(\[Theta])\)\)\/U\&_\^2 - \(3\ \[ImaginaryI]\ dpdx\ H\&_\)\/\(\[Alpha]\ \[Rho]\ U\&_\^2\)\)], "Output"] }, Open ]], Cell["We can recover the Froude number, F.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(pf8 = pf7 /. g -> \ F\ U\&_\^2/H\&_\)], "Input"], Cell[BoxData[ \(TraditionalForm \`\(-\[CapitalGamma]\) + F\ \(sin(\[Theta])\) - \(3\ \[ImaginaryI]\ dpdx\ H\&_\)\/\(\[Alpha]\ \[Rho]\ U\&_\^2\)\)], "Output"] }, Open ]], Cell["\<\ We can eliminate the pressure drop in terms of the Reynolds number. \ First we substitute for the pressure drop for turbulent flow. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(turbpf9 = pf8 /. dpdx \[Rule] \(-\ 0.079\)\ \(\((H\&_\ \[Rho]\ U\&_/\[Mu])\)\^\(- .25\)\) \[Rho]\ U\&_\^2/H\&_\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{\(-\[CapitalGamma]\), "+", \(F\ \(sin(\[Theta])\)\), "+", FractionBox[\(0.23700000000000001`\ \[ImaginaryI]\), RowBox[{"\[Alpha]", " ", SuperscriptBox[\((\(\[Rho]\ H\&_\ U\&_\)\/\[Mu])\), StyleBox["0.25`", StyleBoxAutoDelete->True, PrintPrecision->2]]}]]}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(turbpf10 = turbpf9 /. U\&_ \[Rule] \(R\ \[Mu]\)\/\(\[Rho]\ H\&_\)\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{\(-\[CapitalGamma]\), "+", \(F\ \(sin(\[Theta])\)\), "+", FractionBox[\(0.23700000000000001`\ \[ImaginaryI]\), RowBox[{ SuperscriptBox["R", StyleBox["0.25`", StyleBoxAutoDelete->True, PrintPrecision->2]], " ", "\[Alpha]"}]]}], TraditionalForm]], "Output"] }, Open ]], Cell[TextData["We would like to know the value of \[CapitalGamma]."], "Text"], Cell[CellGroupData[{ Cell["turbgamma=1", "Input"], Cell[BoxData[ \(TraditionalForm\`1\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "The final answer for the pressure variation is", "Subsubsection"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(pressureintturb = turbpf10 /. \[CapitalGamma] -> turbgamma\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{\(F\ \(sin(\[Theta])\)\), "+", FractionBox[\(0.23700000000000001`\ \[ImaginaryI]\), RowBox[{ SuperscriptBox["R", StyleBox["0.25`", StyleBoxAutoDelete->True, PrintPrecision->2]], " ", "\[Alpha]"}]], "-", "1"}], TraditionalForm]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "The final answer for the stress variation is", "Subsubsection"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(stressintturb = \(-\((\((\[Tau]\&^ /. turbtauans[\([1]\)])\) /. U\&_ \[Rule] \(R\ \[Mu]\)\/\(\[Rho]\ H\&_\))\)\) H\&_/h\&^\)], "Input"], Cell[BoxData[ \(TraditionalForm\`0.0395000000000000017`\ \((1\/R)\)\^0.25`\)], "Output"] }, Open ]] }, Open ]] }, Closed]] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Results", "Subtitle"], Cell[CellGroupData[{ Cell["The analytical expressions", "Subsection"], Cell["\<\ You can just click on this section and load in all of the results \ without rederiving them above. Of course, this means that if you change \ something above, you do not want to click on the same result below because it \ will get changed to the old result!!\ \>", "Text"], Cell[CellGroupData[{ Cell["laminar stress", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"lamavestress", "=", FormBox[\(6\/R\), "TraditionalForm"]}]], "Input"], Cell[BoxData[ \(TraditionalForm\`6\/R\)], "Output"] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"lamstress", "=", FormBox[ \(\(-\(\(\[ImaginaryI]\ \((17\ R\^3 - 53900\ R)\)\ \[Alpha]\^2\)\/\(943250\ R\)\)\) + \(4\ \((R\^2 + 1078)\)\ \[Alpha]\)\/\(2695\ R\) + \(6\ \[ImaginaryI]\ \[Alpha]\)\/35 + 12\/R\), "TraditionalForm"]}], ";"}]], "Input", CellTags->"lamstress"] }, Closed]], Cell[TextData[{ ButtonBox["Back", ButtonData:>"conclusions", ButtonStyle->"Hyperlink"], " to conclusions" }], "Text"], Cell[CellGroupData[{ Cell["integral laminar stress", "Subsubsection"], Cell[BoxData[ \(\(stressintlam = 12/R; \)\)], "Input", CellTags->"integral stress"] }, Closed]], Cell[TextData[{ ButtonBox["Back", ButtonData:>"conclusions", ButtonStyle->"Hyperlink"], " to conclusions." }], "Text"], Cell[CellGroupData[{ Cell["laminar pressure", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(lampressure\)], "Input", CellLabel->"In[171]:=", Evaluatable->False], Cell[BoxData[ \(TraditionalForm \`\(3\ \((411\ R\^3 - 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840\ \((\(-1\))\)\^\(4\ m\))\)\ s)\)\ m\^3 + \((\(-3\)\ \((1136 - 2177\ \((\(-1\))\)\^\(2\ m\) + 1225\ \((\(-1\))\)\^\(4\ m\))\)\ s\^2 + \((6531 - 13753\ \((\(-1\))\)\^\(2\ m\) + 7350\ \((\(-1\))\)\^\(4\ m\))\)\ s + 7222\ \((\(-1\))\)\^\(2\ m\) - 3675\ \((\(-1\))\)\^\(4\ m\) - 3547)\)\ m\^2 + \((\(-840\)\ \((\(-1\) + \((\(-1\))\)\^\(2\ m\)) \)\^2 - 3\ \(( 1057 - 1425\ \((\(-1\))\)\^\(2\ m\) + 280\ \((\(-1\))\)\^\(4\ m\))\)\ s\^2 + \((4147 - 5955\ \((\(-1\))\)\^\(2\ m\) + 1680\ \((\(-1\))\)\^\(4\ m\))\)\ s) \)\ m + 120\ s\ \((\((\(-5\) + 7\ \((\(-1\))\)\^\(2\ m\))\)\ s - 7\ \((\(-1\))\)\^\(2\ m\) + 7)\)) \)\ \[Alpha])\)/ \((\((m + 1)\)\ \((m + 2)\)\ \((2\ m + 3)\)\ \((2\ m + 5)\))\))\))\)/ \((35\ R\ \((\(-8\)\ m\^2 - 4\ \((s - 2)\)\ m - 3\ \((\(-1\))\)\^\(2\ m\) + 3\ \((\(-1\))\)\^\(2\ m\)\ s + s + 3)\)\^2)\)\), "TraditionalForm"]}], ";"}]], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["polynomial pressure", "Subsubsection"], Cell[BoxData[ \(\(totalturbpressure; \)\)], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"totalturbpressure", "=", FormBox[ \(\(-\(\(( 144\ \((m - 1)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\)\ \((\(\((\(-1\))\)\^\(2\ m\)\ \((1 - s)\)\)\/\(m - 1\) + \(s - m\)\/\(m - 1\) + 1)\)\ \((2\ m + 1)\)\^2)\)/ \((\((\(-8\)\ m\^2 - 4\ \((s - 2)\)\ m - 3\ \((\(-1\))\)\^\(2\ m\) + 3\ \((\(-1\))\)\^\(2\ m\)\ s + s + 3)\)\ \((8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3)\))\)\)\) - \(144\ \[ImaginaryI]\ \((\((\(-1\))\)\^\(2\ m\)\ \((s - 1)\)\ m + m - s)\)\ \((2\ m + 1)\)\)\/\(R\ \((3\ \((\(-1\))\)\^\(2\ m\)\ \((s - 1)\) + s - 4\ m\ \((2\ m + s - 2)\) + 3)\)\ \[Alpha]\) - \(1\/R\) \((\[ImaginaryI]\ \((\(-\(\(( 144\ \[ImaginaryI]\ \((\(-1\))\)\^\(2\ m - 3\)\ m \ \((2\ m - 2)\)\ \((2\ m - 1)\)\ \((4\ m + 7)\)\ R\ \((s - 1)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\)\ \((2\ m + 1)\)\^2)\)/ \((\((m + 1)\)\ \((m + 2)\)\ \((2\ m + 3)\)\ \((2\ m + 5)\)\ \((8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3) \)\^2)\)\)\) - \((216\ \[ImaginaryI]\ \((\(-1\))\)\^\(2\ m - 2\)\ m\ \((2\ m - 1)\)\ \((16\ m\^3 - 64\ m\^2 - 190\ m - 65)\)\ R\ \((s - 1)\)\ \((\(-\((\(-1\))\)\^\(2\ m\)\)\ m + \((\(-1\))\)\^\(2\ m\)\ s\ m + m - s)\)\ \((2\ m + 1)\)\^2)\)/ \((\((m + 1)\)\ \((m + 2)\)\ \((2\ m + 3)\)\ \((2\ m + 5)\)\ \((4\ m + 2)\)\ \((8\ m\^2 + 4\ s\ m - 8\ m + 3\ \((\(-1\))\)\^\(2\ m\) - 3\ \((\(-1\))\)\^\(2\ m\)\ s - s - 3)\)\^2) \) - \(( 9\ \[ImaginaryI]\ R\ \((768\ \((\((\(-1\))\)\^\(2\ m\)\ s - \((\(-1\))\)\^\(2\ m\) + 1)\)\ m\^7 + 64\ \(( 65\ \((\(-1\))\)\^\(2\ m\)\ s\^2 - 52\ \((\(-1\))\)\^\(2\ m\)\ s + 53\ s - 13\ \((\(-1\))\)\^\(2\ m\) + 13)\)\ m\^6 + 32\ \(( 65\ \(( \(-2\) + 15\ \((\(-1\))\)\^\(2\ m\)) \)\ s\^2 + \((949 - 1620\ \((\(-1\))\)\^\(2\ m\)) \)\ s + 645\ \((\(-1\) + \((\(-1\))\)\^\(2\ m\)) \))\)\ m\^5 + 80\ \(( \((\(-390\) + 1105\ \((\(-1\))\)\^\(2\ m\) + 63\ \((\(-1\))\)\^\(4\ m\))\)\ s\^2 + \((1363 - 2081\ \((\(-1\))\)\^\(2\ m\) - 126\ \((\(-1\))\)\^\(4\ m\))\)\ s + 976\ \((\(-1\))\)\^\(2\ m\) + 63\ \((\(-1\))\)\^\(4\ m\) - 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1\), "TraditionalForm"]}], ";"}]], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Comparison of laminar differential and integral cases.", "Subsection", CellTags->"lamdifcomparison"], Cell["\<\ The first term of the pressure and shear stress for each of the two \ cases are identical. The second term in the pressure fluctuation is about \ 20% different. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(N[\((54/35 - 6/5)\)/\((54/35)\)]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`0.222222222222222232`\)], "Output"] }, Open ]], Cell[TextData[ "There is no second term in the shear stress for the integral model. A \ normalization of the second term of the second term is 6/35 \[Alpha]/(6/R) = \ fluctation/average stress. It could be that this is less important that the \ pressure in causing wave growth, but this difference could be important. 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The laminar and polynomial differential results have \ more significant qualitative differences in both the ", ButtonBox["pressure", ButtonData:>"lampoly press comp", ButtonStyle->"Hyperlink"], " and ", ButtonBox["shear stress", ButtonData:>"lampoly stress comp", ButtonStyle->"Hyperlink"], ". Note that these comparisons are made after normalization by the average \ stress-- which is different for laminar versus turbulent flow. \n\n3. The \ polynomial and turbulent integral models give very different answers for the \ ", ButtonBox["pressure", ButtonData:>"turb pressure comps", ButtonStyle->"Hyperlink"], " and ", ButtonBox["shear stress", ButtonData:>"turb stress comp", ButtonStyle->"Hyperlink"], ". This is troubling because even though the polynomial model is not \ necessarily correct, it produces the same average pressure drop as the \ integral model. In the absence of a better differential model that reliably \ produces the correct basestates and has been verified to produce reasonable \ growth information, we cannot completely discount either of these two. Thus, \ at the present time,", StyleBox[ " there is a great deal of uncertainty in the prediction of the stability \ behavior for turbulent flows", FontVariations->{"Underline"->True}], " -- the only case that matters for process conditions. \n\n4. Finally, \ we have shown above how to solve a difficult ordinary differential equation \ by a regular perturbation method. Note that presence of inhomogeneities in \ the boundary conditions at the lowest order from d", StyleBox["omain perturbation", FontSlant->"Italic"], " that allows us to evaluate the interfacial conditions (very much more \ conveniently) at y=0, instead of the exact position of the interface. 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