Solving Navier-Stokes equations for a steady-state compressible viscous flow in a 2D axisymmetric step

Question

Note: you may apply or follow the edits on the code here in this GitHub Gist

I'm trying to follow this post to solve Navier-Stokes equations for a compressible viscous flow in a 2D axisymmetric step. The geometry is :

lc = 0.03;
rc = 0.01;
xp = 0.01;
c = 0.005;
rp = rc - c;
lp = lc - xp;
Subscript[T, 0] = 300;
Subscript[\[Eta], 0] = 1.846*10^-5;
Subscript[P, 1] = 6*10^5 ;
Subscript[P, 0] = 10^5;
Subscript[c, P] = 1004.9;
Subscript[c, \[Nu]] = 717.8;
Subscript[R, 0] = Subscript[c, P] - Subscript[c, \[Nu]];
\[CapitalOmega] = RegionDifference[
   Rectangle[{0, 0}, {lc, rc}], 
   Rectangle[{xp, 0}, {xp + lp, rp}]];

And meshing:

Needs["NDSolve`FEM`"];
mesh = ToElementMesh[\[CapitalOmega], 
   "MaxBoundaryCellMeasure" -> 0.00001, 
   MaxCellMeasure -> {"Length" -> 0.0008}, 
   "MeshElementConstraint" -> 20, MeshQualityGoal -> "Maximal"][
  "Wireframe"]

Where the model is axisymmetric around the x axis, the governing equations including conservation equations of mass, momentum and heat can be written as:

$$ \frac{\partial}{\partial x}\left( \rho \nu_x \right)+\frac{1}{r}\frac{\partial }{\partial r}\left(r \rho \nu_r\right)=0 \tag{1}$$

$$\frac{\partial}{\partial x}\left( \rho \nu_x^2+\mathring{R} \rho T \right)+\frac{1}{r}\frac{\partial}{\partial r}\left( r \left( \rho \nu_r \nu_x + \eta \frac{\partial \nu_x}{\partial r} \right)\right) \tag{2}$$

$$ \frac{\partial}{\partial x}\left( \rho \nu_x \nu_r+\eta \frac{\partial \nu_r}{\partial x} \right)+ \frac{1}{r}\frac{\partial}{\partial r}\left( r \left( \rho \nu_r ^2 +\mathring{R} \rho T \right) \right)=0 \tag{3}$$

$$\rho c_\nu\left( \nu_x \frac{\partial T}{\partial x} + \nu_r \frac{\partial T}{\partial r} \right)+ \mathring{R} \rho T \left( \frac{1}{r}\frac{\partial}{\partial r} \left( r \nu_r \right)+ \frac{\partial \nu_x}{\partial x} \right)+ \eta \left( 2 \left( \frac{\partial \nu_x}{\partial x} \right)^2+ 2 \left( \frac{\partial \nu_r}{\partial r} \right)^2+ \left( \frac{\partial \nu_r}{\partial x}+ \frac{\partial \nu_x}{\partial r} \right)^2 \\ -\frac{2}{3}\left( \frac{1}{r} \frac{\partial}{\partial r}\left( r \nu_r \right) + \frac{\partial \nu_x}{\partial x} \right)^2 \right)=0 \tag{4}$$

eqn1 = D[\[Rho][x, r]*Subscript[\[Nu], x][x, r], x] + 
    D[r*\[Rho][x, r]*Subscript[\[Nu], r][x, r], r]/r == 0 ;
eqn2 = D[\[Rho][x, r]*Subscript[\[Nu], x][x, r]^2 + 
      Subscript[R, 0] \[Rho][x, r]*T[x, r], x] + 
    D[r*(\[Rho][x, r]*Subscript[\[Nu], x][x, r]*
          Subscript[\[Nu], r][x, r] + 
         Subscript[\[Eta], 0]*D[Subscript[\[Nu], x][x, r], r]), r]/
     r == 0 ;
eqn3 = D[\[Rho][x, r]*Subscript[\[Nu], x][x, r]*
       Subscript[\[Nu], r][x, r] + 
      Subscript[\[Eta], 0]*D[Subscript[\[Nu], r][x, r], x], x] + 
    D[r*(\[Rho][x, r]*Subscript[\[Nu], r][x, r]^2 + 
         Subscript[R, 0] \[Rho][x, r]*T[x, r]), r]/r == 0;
eqn4 = Subscript[
     c, \[Nu]]*\[Rho][x, 
      r]*(Subscript[\[Nu], x][x, r]*D[T[x, r], x] + 
       Subscript[\[Nu], r][x, r]*D[T[x, r], r]) + 
    Subscript[R, 0]*\[Rho][x, r]*
     T[x, r]*(D[Subscript[\[Nu], x][x, r], x] + 
       D[r*Subscript[\[Nu], r][x, r], x]/r) + (2*
        D[Subscript[\[Nu], x][x, r], x]^2 + 
       2*D[Subscript[\[Nu], r][x, r], 
          r]^2 + (D[Subscript[\[Nu], x][x, r], r] + 

          D[Subscript[\[Nu], r][x, r], 
           x])^2 - ((D[Subscript[\[Nu], x][x, r], x] + 
            D[r*Subscript[\[Nu], r][x, r], x]/r)^2)*2/3)*
     Subscript[\[Eta], 0] == 0;
eqns = {eqn1, eqn2, eqn3, eqn4};

And the boundary conditions are:

constant pressure at inlet
constant pressure at outlet
axis of symmetry
no slip

Implemented as

bc1 = Subscript[R, 0] \[Rho][0, r]*Subscript[T, 0] == Subscript[P, 1] 
bc2 = Subscript[R, 0] \[Rho][lc, r]*Subscript[T, 0] == Subscript[P, 0]
bc3 = DirichletCondition[{Subscript[\[Nu], r][x, 0] == 0, 
   D[Subscript[\[Nu], r][x, r], r] == 0, 
   D[Subscript[\[Nu], x][x, r], r] == 0, D[\[Rho][x, r], r] == 0, 
   D[T[x, r], r] == 0}, r == 0 && (0 <= x <= xp  )] 
bc4 = DirichletCondition[{Subscript[\[Nu], r][x, r] == 0, 
    Subscript[\[Nu], x][x, r] == 
     0}, (0 <= r <= rp && x == xp ) || (r == rp && 
      xp <= x <= xp + lp)  || (r == rc && 0 <= x <= lc) ] == 0
bcs = {bc1, bc2, bc3, bc4};

When I try to solve the equations:

{\[Nu]xsol, \[Nu]rsol, \[Rho]sol, Tsol} = 
  NDSolveValue[{eqns, , bcs}, {Subscript[\[Nu], x], Subscript[\[Nu], 
    r], \[Rho], T}, {x, r} \[Element] mesh, 
   Method -> {"FiniteElement", 
     "InterpolationOrder" -> {Subscript[\[Nu], x] -> 2, 
       Subscript[\[Nu], r] -> 2, \[Rho] -> 1, T -> 1}, 
     "IntegrationOrder" -> 5}];

I get the errors:

NDSolveValue::femnr: {x,r}[Element] is not a valid region specification.

and

Set::shape: Lists {[Nu]xsol,[Nu]rsol,Tsol,[Rho]sol} and NDSolveValue[<<1>>] are not the same shape.

Googling the errors does not offer that much of help (e.g. here). I would appreciate if you could help me know What is the issue and how I can solve it.

P.S.1. The NDSolveValue femnr error was caused by [ "Wireframe"] term at the end of meshing command changing it to

mesh = ToElementMesh[\[CapitalOmega], 
   "MaxBoundaryCellMeasure" -> 0.00001, 
   MaxCellMeasure -> {"Length" -> 0.0008}, 
   "MeshElementConstraint" -> 20, MeshQualityGoal -> "Maximal"];
mesh["Wireframe"]

resolves the issue.

P.S.2. There is an extra ==0 at the end of boundary condition 4 it was edited to:

bc4 = DirichletCondition[{Subscript[\[Nu], r][x, r] == 0, 
    Subscript[\[Nu], x][x, r] == 
     0}, (0 <= r <= rp && x == xp) || (r == rp && 
      xp <= x <= xp + lp) || (r == rc && 0 <= x <= lc)];

at this moment the second error still persists and a new error was added:

NDSolveValue::deqn Equation or list of equations expected instead of Null in the first argument ...

P.S.3 There were multiple issues. So I decided to use this Github Gist to further edit the code.

Start by removing ["Wireframe"] in the definition of mesh. You have not defined eqns: maybe eqns = {eqn1, eqn2, eqn3, eqn4}? — anderstood, Commented Feb 21, 2018 at 15:32
Now I get NDSolveValue::overdet: There are fewer dependent variables, {T[x,r],\[Rho][x,r]}, than equations, so the system is overdetermined.. Maybe the Subscript pose problem in the name of the variables (it seems nux and nur are not recognized as unknowns)? — anderstood, Commented Feb 21, 2018 at 15:52
The error NDSolveValue::deqn Equation or list of equations expected instead of Null in the first argument ... is due to the two successive commas in NDSolveValue[{eqns, , bcs},. — anderstood, Commented Feb 21, 2018 at 15:54
What may or may not be tricky is the linearization. You could start with a simpler version as outlines in the other post you reference. (u.del(u))^k+1 approx= u^k.del(u)^k+1. I'd be curious to see the result. Good luck. — user21, Commented Feb 22, 2018 at 10:23
Also, it might be useful to add a note that your equations are not quite right. This save the time to try to solve them. — user21, Commented Apr 17, 2019 at 8:39

Alex Trounev · Accepted Answer · 2018-08-13 15:52:40Z

I prepared a solver for the steady axisymmetric flow of a viscous incompressible fluid for Reynolds numbers up to 1000 (but it is possible more, up to loss of stability) in geometry, which the author discusses. After discussing my decision, we will move on to the compressible flow, but I do not promise that it will be fast. For the solution, I used a standard task, which was published in the documentation since version 11. I even saved the notation for the solution to be recognizable. To solve a nonlinear problem, I used the fixed-point method. Here is an example of solving a problem with a Reynolds number of 1000. If you want to increase this number, you must decrease the parameter MaxCellMeasure.

lc = 3;
rc = 1;
xp = 1;
c = .5;
rp = rc - c;
lp = lc - xp;
K = 25; Re0 = 1000; h = .001;
u0[y_] := (rc^2 - y^2)/2;

\[CapitalOmega] = 
  RegionDifference[Rectangle[{0, h}, {lc, rc}], 
   Rectangle[{xp, rp}, {lc, rc}]];
UX[0][x_, y_] := 0;
VY[0][x_, y_] := 0;
Do[
  {UX[i], VY[i]} = 
   NDSolveValue[{{Inactive[
           Div][({{-\[Mu], 0}, {0, -\[Mu]}}.Inactive[Grad][
             u[x, y], {x, y}]), {x, y}] - D[u[x, y], y]*y/(y^2 + 0.) + 
\!\(\*SuperscriptBox[\(p\), 
TagBox[
RowBox[{"(", 
RowBox[{"1", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y] + Re0*UX[i - 1][x, y]*D[u[x, y], x] +
          Re0*VY[i - 1][x, y]*D[u[x, y], y], 
        Inactive[
           Div][({{-\[Mu], 0}, {0, -\[Mu]}}.Inactive[Grad][
             v[x, y], {x, y}]), {x, y}] + v[x, y]/(y^2 + 0.) - 
         D[v[x, y], y]*y/(y^2 + 0.) + 
\!\(\*SuperscriptBox[\(p\), 
TagBox[
RowBox[{"(", 
RowBox[{"0", ",", "1"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y] + Re0*UX[i - 1][x, y]*D[v[x, y], x] +
          Re0*VY[i - 1][x, y]*D[v[x, y], y], 
        D[y*u[x, y], x] + D[y*v[x, y], y]} == {0, 0, 0} /. \[Mu] -> 
       1, {
      DirichletCondition[u[x, y] == u0[y], x == 0.], 
      DirichletCondition[{u[x, y] == 0., v[x, y] == 0.}, 
       0 <= x <= lc && y == rc || y == rp], 
      DirichletCondition[{u[x, y] == 0., v[x, y] == 0.}, 
       x == xp && rp <= y <= rc],
      DirichletCondition[p[x, y] == 0, x == lc], 
      DirichletCondition[v[x, y] == 0, y == h]}}, {u, 
     v}, {x, y} \[Element] \[CapitalOmega], 
    Method -> {"FiniteElement", 
      "InterpolationOrder" -> {u -> 2, v -> 2, p -> 1}, 
      "MeshOptions" -> {"MaxCellMeasure" -> 0.0005}}], {i, 1, K}];
StreamDensityPlot[{UX[K][x, y], 
  VY[K][x, y]}, {x, y} \[Element] \[CapitalOmega], 
 StreamPoints -> Fine, StreamStyle -> LightGray, 
 PlotLegends -> Automatic, VectorPoints -> Fine, 
 ColorFunction -> "TemperatureMap", FrameLabel -> {"x", "y"}]

{ContourPlot[UX[K][x, y], {x, y} \[Element] \[CapitalOmega], 
  PlotLegends -> Automatic, Contours -> 20, PlotPoints -> 25, 
  ColorFunction -> "TemperatureMap", AxesLabel -> {"x", "y"}, 
  PlotLabel -> u], 
 ContourPlot[VY[K][x, y], {x, y} \[Element] \[CapitalOmega], 
  PlotLegends -> Automatic, Contours -> 20, PlotRange -> All, 
  PlotPoints -> 50, ColorFunction -> "TemperatureMap", 
  AxesLabel -> {"x", "y"}, PlotLabel -> v], 
 ContourPlot[
  Norm[{UX[K][x, y], VY[K][x, y]}], {x, y} \[Element] \[CapitalOmega],
   PlotLegends -> Automatic, Contours -> 20, PlotRange -> All, 
  PlotPoints -> 50, ColorFunction -> "TemperatureMap", 
  AxesLabel -> {"x", "y"}, PlotLabel -> Sqrt[u^2 + v^2]]}

How can we know that the solution converges? Let us consider a simple estimate for the difference of two solutions at neighboring steps:

ListLogPlot[
 Table[Sum[Abs[UX[i][x, xp/2] - UX[i - 1][x, xp/2]], {x, 0, lc, .01}]/
   Sum[1, {x, 0, lc, .01}], {i, 1, K}], Filling -> Axis]

In this example, we see a rapid convergence with increasing K.

It was possible to make a solver for the isentropic flow. Here is an example of a flow with subsonic and supersonic speed in an axisymmetric channel with the geometry proposed by the author. The Reynolds number is 500. The Mach number at the exit from the channel is 2.5. A solver for an incompressible fluid is used with the necessary corrections that take into account the compressibility.

lc = 3;
rc = 1;
xp = 1;
c = .5;
rp = rc - c;
lp = lc - xp;  q = .4;
K = 25; Re0 = 500; h = .001; M = 1.; Re1 = Re0/M^2;
u0[y_] := (rc^2 - y^2)/2;
    \[CapitalOmega] = 
  RegionDifference[Rectangle[{0, h}, {lc, rc}], 
   Rectangle[{xp, rp}, {lc, rc}]];
UX[0][x_, y_] := 0;
VY[0][x_, y_] := 0;
\[CapitalRho][0][x_, y_] := 1;
Do[
  {UX[i], VY[i], \[CapitalRho][i]} = 
   NDSolveValue[{{Inactive[
           Div][({{-\[Mu], 0}, {0, -\[Mu]}}.Inactive[Grad][
             u[x, y], {x, y}]), {x, y}] - D[u[x, y], y]*y/(y^2 + 0.) +
          Re1*(Abs[\[CapitalRho][i - 1][x, y]]^q)*
\!\(\*SuperscriptBox[\(\[Rho]\), 
TagBox[
RowBox[{"(", 
RowBox[{"1", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y] + Re0*UX[i - 1][x, y]*D[u[x, y], x] +
          Re0*VY[i - 1][x, y]*D[u[x, y], y], 
        Inactive[
           Div][({{-\[Mu], 0}, {0, -\[Mu]}}.Inactive[Grad][
             v[x, y], {x, y}]), {x, y}] + v[x, y]/(y^2 + 0.) - 
         D[v[x, y], y]*y/(y^2 + 0.) + 
         Re1*(Abs[\[CapitalRho][i - 1][x, y]^q])*
\!\(\*SuperscriptBox[\(\[Rho]\), 
TagBox[
RowBox[{"(", 
RowBox[{"0", ",", "1"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y] + Re0*UX[i - 1][x, y]*D[v[x, y], x] +
          Re0*VY[i - 1][x, y]*D[v[x, y], y], 
        D[y*\[CapitalRho][i - 1][x, y]*u[x, y], x] + 
         D[y*\[CapitalRho][i - 1][x, y]*v[x, y], y]} == {0, 0, 
        0} /. \[Mu] -> 1, {
      DirichletCondition[{u[x, y] == u0[y]}, x == 0.], 
      DirichletCondition[{u[x, y] == 0., v[x, y] == 0.}, 
       0 <= x <= lc && y == rc || y == rp], 
      DirichletCondition[{u[x, y] == 0., v[x, y] == 0.}, 
       x == xp && rp <= y <= rc],
      DirichletCondition[\[Rho][x, y] == 1, x == lc], 
      DirichletCondition[v[x, y] == 0, y == h]}}, {u, 
     v, \[Rho]}, {x, y} \[Element] \[CapitalOmega], 
    Method -> {"FiniteElement", 
      "InterpolationOrder" -> {u -> 2, v -> 2, \[Rho] -> 1}, 
      "MeshOptions" -> {"MaxCellMeasure" -> 0.0005}}], {i, 1, K}];
{ContourPlot[\[CapitalRho][K][x, 
   y], {x, y} \[Element] \[CapitalOmega], PlotLegends -> Automatic, 
  Contours -> 20, PlotPoints -> 25, ColorFunction -> "TemperatureMap",
   AxesLabel -> {"x", "y"}, PlotLabel -> \[Rho]], 
 ContourPlot[
  Norm[{UX[K][x, y], VY[K][x, y]}]/\[CapitalRho][K][x, y]^(q/2), {x, 
    y} \[Element] \[CapitalOmega], PlotLegends -> Automatic, 
  Contours -> 20, PlotRange -> All, PlotPoints -> 50, 
  ColorFunction -> "TemperatureMap", AxesLabel -> {"x", "y"}, 
  PlotLabel -> "M"]}

Distribution of density and Mach number

Let's add another pair of Fig. for the velocity field and the convergence of the method in the case of a compressible flow.

StreamDensityPlot[{UX[K][x, y], 
  VY[K][x, y]}, {x, y} \[Element] \[CapitalOmega], 
 StreamPoints -> Fine, StreamStyle -> LightGray, 
 PlotLegends -> Automatic, VectorPoints -> Fine, 
 ColorFunction -> "TemperatureMap", FrameLabel -> {"x", "y"}]

ListLogPlot[
 Table[Sum[Abs[UX[i][x, xp/2] - UX[i - 1][x, xp/2]], {x, 0, lc, .01}]/
   Sum[1, {x, 0, lc, .01}], {i, 1, K}], Filling -> Axis]

In the case of a compressible viscous flow with a given pressure at the inlet and outlet, I recommend the following code

lc = 3;
rc = 1;
xp = 1;
c = .5;
rp = rc - c;
lp = lc - xp; q = .4;
K = 17; Re0 = 100; h = .001; M0 = 1; Re1 = Re0/M0^2;
u0[y_] := (rc^2 - y^2)/2;
\[CapitalOmega] = 
  RegionDifference[Rectangle[{0, h}, {lc, rc}], 
   Rectangle[{xp, rp}, {lc, rc}]];
UX[0][x_, y_] := 0;
VY[0][x_, y_] := 0;
\[CapitalRho][0][x_, y_] := 1;
Do[
  {UX[i], VY[i], \[CapitalRho][i]} = 
   NDSolveValue[{{Inactive[
           Div][({{-\[Mu], 0}, {0, -\[Mu]}}.Inactive[Grad][
             u[x, y], {x, y}]), {x, y}] - D[u[x, y], y]/y + 
         Re1*(Abs[\[CapitalRho][i - 1][x, y]]^q)*
\!\(\*SuperscriptBox[\(\[Rho]\), 
TagBox[
RowBox[{"(", 
RowBox[{"1", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y] + Re0*UX[i - 1][x, y]*D[u[x, y], x] +
          Re0*VY[i - 1][x, y]*D[u[x, y], y], 
        Inactive[
           Div][({{-\[Mu], 0}, {0, -\[Mu]}}.Inactive[Grad][
             v[x, y], {x, y}]), {x, y}] + v[x, y]/y^2 - 
         D[v[x, y], y]/y + Re1*(Abs[\[CapitalRho][i - 1][x, y]^q])*
\!\(\*SuperscriptBox[\(\[Rho]\), 
TagBox[
RowBox[{"(", 
RowBox[{"0", ",", "1"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y] + Re0*UX[i - 1][x, y]*D[v[x, y], x] +
          Re0*VY[i - 1][x, y]*D[v[x, y], y], 
        D[y*\[CapitalRho][i - 1][x, y]*u[x, y], x] + 
         D[y*\[CapitalRho][i - 1][x, y]*v[x, y], y]} == {0, 0, 
        0} /. \[Mu] -> 1, {
      DirichletCondition[{\[Rho][x, y] == 2, v[x, y] == 0}, x == 0.], 
      DirichletCondition[{u[x, y] == 0., v[x, y] == 0.}, 
       0 <= x <= lc && y == rc || y == rp], 
      DirichletCondition[{u[x, y] == 0., v[x, y] == 0.}, 
       x == xp && rp <= y <= rc],
      DirichletCondition[{\[Rho][x, y] == 1, v[x, y] == 0}, x == lc], 
      DirichletCondition[v[x, y] == 0, y == h]}}, {u, 
     v, \[Rho]}, {x, y} \[Element] \[CapitalOmega], 
    Method -> {"FiniteElement", 
      "InterpolationOrder" -> {u -> 2, v -> 2, \[Rho] -> 1}, 
      "MeshOptions" -> {"MaxCellMeasure" -> 0.0005}}], {i, 1, K}];
StreamDensityPlot[{UX[K][x, y], 
  VY[K][x, y]}, {x, y} \[Element] \[CapitalOmega], 
 StreamPoints -> Fine, StreamStyle -> LightGray, 
 PlotLegends -> Automatic, VectorPoints -> Fine, 
 ColorFunction -> "TemperatureMap", FrameLabel -> {"x", "y"}]

{ContourPlot[\[CapitalRho][K][x, 
   y], {x, y} \[Element] \[CapitalOmega], PlotLegends -> Automatic, 
  Contours -> 20, PlotPoints -> 25, ColorFunction -> "TemperatureMap",
   AxesLabel -> {"x", "y"}, PlotLabel -> \[Rho]], 
 ContourPlot[
  Norm[{UX[K][x, y], VY[K][x, y]}]/\[CapitalRho][K][x, y]^(q/2), {x, 
    y} \[Element] \[CapitalOmega], PlotLegends -> Automatic, 
  Contours -> 20, PlotRange -> All, PlotPoints -> 50, 
  ColorFunction -> "TemperatureMap", AxesLabel -> {"x", "y"}, 
  PlotLabel -> "M"]}

In Fig. The distributions of the velocity, density and Mach number

thanks alot. I will go through your code and come back here. — Foad, Commented Aug 10, 2018 at 12:34
I updated the code with compressibility in the case of isentropic flow — Alex Trounev, Commented Aug 12, 2018 at 7:15
I showed the method of solution and the working code. You can put any conditions. Typically, for steady flows, the gas flow at the inlet and outlet pressure are set. — Alex Trounev, Commented Aug 13, 2018 at 11:42
I checked that the boundary conditions with pressure work. See update. — Alex Trounev, Commented Aug 13, 2018 at 15:45
In turn, can you give a link to the source of your equations? This is more like Prandtl's equations than the Navier-Stokes equations. Open the Help in Mathematica, look for the Stokes operator on the page FEMDocumentation/tutorial/SolvingPDEwithFEM. There is a basic example for solving the system of equations for a viscous incompressible fluid at zero Reynolds number in a channel with a discontinuity. Add to this system a cylindrical symmetry and a nonzero Reynolds number. You'll get my first example. — Alex Trounev, Commented Aug 14, 2018 at 3:01

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