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3D stable fluids transition simulation algorithm: laminar to turbulent flow

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Using exact benchmark solution we also can test linear FEM algorithm described on Solver for unsteady flow with the use of Mathematica FEM and extended to 3D below as follows

Needs["NDSolve`FEM`"]
reg = Cuboid[];
mesh = ToElementMesh[reg, MaxCellMeasure -> 0.001]

U[x_, y_, z_, 
  t_] := -a Exp[-d^2 t] (Exp[a x] Sin[a y + d z] + 
    Exp[a z] Cos[a x + d y]); 
V[x_, y_, z_, 
  t_] := -a Exp[-d^2 t] (Exp[a y] Sin[a z + d x] + 
    Exp[a x] Cos[a y + d z]); 
W[x_, y_, z_, 
  t_] := -a Exp[-d^2 t] (Exp[a z] Sin[a x + d y] + 
    Exp[a y] Cos[a z + d x]);
P[x_, y_, z_, 
  t_] := -a ^2/
   2 Exp[-2 d^2 t] (Exp[2 a x] + Exp[2 a y] + Exp[2 a z] + 
    2 Sin[a x + d y] Exp[a (y + z)] Cos[a z + d x] + 
    2 Sin[a y + d z] Exp[a (x + z)] Cos[a x + d y] + 
    2 Sin[a z + d x] Exp[a (y + x)] Cos[
      a y + d z]); a = 1; d = 1; t0 = 1/400; nn = 200; \[Nu] = 1;
UX[0] = U[x, y, z, 0];
VY[0] = V[x, y, z, 0]; WZ[0] = W[x, y, z, 0];
P0[0] = P[x, y, z, 0];
Do[
   {UX[i], VY[i], WZ[i], P0[i]} = 
     NDSolveValue[{{-\[Nu]*
           Laplacian[
            u[x, y, z], {x, y, z}] + (u[x, y, z] - UX[i - 1])/t0 + 
          UX[i - 1]*D[u[x, y, z], x] + VY[i - 1]*D[u[x, y, z], y] + 
          WZ[i - 1]*D[u[x, y, z], z] + 
\!\(\*SuperscriptBox[\(p\), 
TagBox[
RowBox[{"(", 
RowBox[{"1", ",", "0", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y, z], -\[Nu]*
           Laplacian[
            v[x, y, z], {x, y, z}] + (v[x, y, z] - VY[i - 1])/t0 + 
          UX[i - 1]*D[v[x, y, z], x] + VY[i - 1]*D[v[x, y, z], y] + 
          WZ[i - 1]*D[v[x, y, z], z] + 
\!\(\*SuperscriptBox[\(p\), 
TagBox[
RowBox[{"(", 
RowBox[{"0", ",", "1", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y, z], -\[Nu]*
           Laplacian[
            w[x, y, z], {x, y, z}] + (w[x, y, z] - WZ[i - 1])/t0 + 
          UX[i - 1]*D[w[x, y, z], x] + VY[i - 1]*D[w[x, y, z], y] + 
          WZ[i - 1]*D[w[x, y, z], z] + 
\!\(\*SuperscriptBox[\(p\), 
TagBox[
RowBox[{"(", 
RowBox[{"0", ",", "0", ",", "1"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y, z], (
\!\(\*SuperscriptBox[\(u\), 
TagBox[
RowBox[{"(", 
RowBox[{"1", ",", "0", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y, z] + 
\!\(\*SuperscriptBox[\(v\), 
TagBox[
RowBox[{"(", 
RowBox[{"0", ",", "1", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y, z] + 
\!\(\*SuperscriptBox[\(w\), 
TagBox[
RowBox[{"(", 
RowBox[{"0", ",", "0", ",", "1"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y, z])} == {0, 0, 0, 0}, {
        DirichletCondition[{u[x, y, z] == U[x, y, z, i t0], 
          v[x, y, z] == V[x, y, z, i t0], 
          w[x, y, z] == W[x, y, z, i t0], 
          p[x, y, z] == P[x, y, z, i t0]}, True]}}, {u[x, y, z] , 
       v[x, y, z] , w[x, y, z] , p[x, y, z]}, {x, y, z} \[Element] 
       mesh, Method -> {"FiniteElement", 
        "InterpolationOrder" -> {u -> 2, v -> 2, w -> 2, 
          p -> 1}}];, {i, 1, nn}]; 

Visualization of error on every 5 steps on the line $y=1/2,z=1/2$ for $u_n=(UX[n],VY[n],WZ[n])$, and $p=P0[n]$

Table[{k t0 // N, 
  Plot[Evaluate[{U[x, y, z, k t0]/UX[k] - 1} /. {y -> 1/2, 
      z -> 1/2}], {x, 0, 1}, PlotRange -> All], 
  Plot[Evaluate[{V[x, y, z, k t0]/VY[k] - 1} /. {y -> 1/2, 
      z -> 1/2}], {x, 0, 1}, PlotRange -> All], 
  Plot[Evaluate[{W[x, y, z, k t0]/WZ[k] - 1} /. {y -> 1/2, 
      z -> 1/2}], {x, 0, 1}, PlotRange -> All], 
  Plot[Evaluate[{P[x, y, z, k t0]/P0[k] - 1} /. {y -> 1/2, 
      z -> 1/2}], {x, 0, 1}, PlotRange -> All]}, {k, 5, nn, 5}]

Note, that in the picture shown last steps only. The relative error for pressure is about $2×10^{−3}$, and for velocity $1.5×10^{−4}$ on the grid 10×10×10 on 200 steps in time with time step τ=1/400. Figure 1

To test projection step itself we can use Mathematica FEM and exact benchmark solution from well known paper EXACT FULLY 3D NAVIER-STOKES SOLUTIONS FOR BENCHMARKING by C. ROSS ETHIER AND D. A. STEINMAN as follows. First, we consider projection step as an implementation of the predictor-corrector algorithm $$\frac{u-u_n}{\tau}+(u.\nabla)u-\nu\nabla^2 u=0$$ $$\frac{u_{n+1}-u}{\tau}+\nabla p =0$$ here $\tau$ is time step, $u_n, u, u_{n+1}$ if velocity field on previous, intermediate and next step consequently, and $p$ is a pressure. We suppose that $\nabla.u_{n+1}=0$, and therefore $$\nabla^2p-\frac{\nabla.u}{\tau}=0$$ Second, we solve NSE in the unit cuboid with Dirichlet condition using exact solution and FEM, we have code

Clear["Global`*"]
Needs["NDSolve`FEM`"]
reg = Cuboid[];   
mesh = ToElementMesh[reg, MaxCellMeasure -> 0.001];
(*Exact solution*)
U[x_, y_, z_, 
  t_] := -a Exp[-d^2 t] (Exp[a x] Sin[a y + d z] + 
    Exp[a z] Cos[a x + d y]); 
V[x_, y_, z_, 
  t_] := -a Exp[-d^2 t] (Exp[a y] Sin[a z + d x] + 
    Exp[a x] Cos[a y + d z]); 
W[x_, y_, z_, 
  t_] := -a Exp[-d^2 t] (Exp[a z] Sin[a x + d y] + 
    Exp[a y] Cos[a z + d x]);
P[x_, y_, z_, 
  t_] := -a ^2/
   2 Exp[-2 d^2 t] (Exp[2 a x] + Exp[2 a y] + Exp[2 a z] + 
    2 Sin[a x + d y] Exp[a (y + z)] Cos[a z + d x] + 
    2 Sin[a y + d z] Exp[a (x + z)] Cos[a x + d y] + 
    2 Sin[a z + d x] Exp[a (y + x)] Cos[
      a y + d z]); 
(*t0 is time step, nn is number of iterations *)
a = 1; d = 1; t0 = 1/400; nn = 200; \[Nu] = 1;
(*FEM implementation of predictor-corrector algorithm*)
UX[0] = U[x, y, z, 0];
VY[0] = V[x, y, z, 0]; WZ[0] = W[x, y, z, 0];
P0[0] = P[x, y, z, 0];
Do[
   {UX[i], VY[i], WZ[i], P0[i]} = 
     NDSolveValue[{{-\[Nu]*
           Laplacian[
            u[x, y, z], {x, y, z}] + (u[x, y, z] - UX[i - 1])/t0 + 
          UX[i - 1]*D[u[x, y, z], x] + VY[i - 1]*D[u[x, y, z], y] + 
          WZ[i - 1]*D[u[x, y, z], z], -\[Nu]*
           Laplacian[
            v[x, y, z], {x, y, z}] + (v[x, y, z] - VY[i - 1])/t0 + 
          UX[i - 1]*D[v[x, y, z], x] + VY[i - 1]*D[v[x, y, z], y] + 
          WZ[i - 1]*D[v[x, y, z], z], -\[Nu]*
           Laplacian[
            w[x, y, z], {x, y, z}] + (w[x, y, z] - WZ[i - 1])/t0 + 
          UX[i - 1]*D[w[x, y, z], x] + VY[i - 1]*D[w[x, y, z], y] + 
          WZ[i - 1]*D[w[x, y, z], z], 
         Laplacian[p[x, y, z], {x, y, z}] - (
\!\(\*SuperscriptBox[\(u\), 
TagBox[
RowBox[{"(", 
RowBox[{"1", ",", "0", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y, z] + 
\!\(\*SuperscriptBox[\(v\), 
TagBox[
RowBox[{"(", 
RowBox[{"0", ",", "1", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y, z] + 
\!\(\*SuperscriptBox[\(w\), 
TagBox[
RowBox[{"(", 
RowBox[{"0", ",", "0", ",", "1"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y, z])/t0} == {0, 0, 0, 0}, {
        DirichletCondition[{u[x, y, z] == 
           U[x, y, z, i t0 - t0/2] + t0 D[P[x, y, z, i t0], x], 
          v[x, y, z] == 
           V[x, y, z, i t0 - t0/2] + t0 D[P[x, y, z, i t0], y], 
          w[x, y, z] == 
           W[x, y, z, i t0 - t0/2] + t0 D[P[x, y, z, i t0], z], 
          p[x, y, z] == P[x, y, z, i t0 ]}, True]}}, {u[x, y, z] - t0 
\!\(\*SuperscriptBox[\(p\), 
TagBox[
RowBox[{"(", 
RowBox[{"1", ",", "0", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y, z], v[x, y, z] - t0 
\!\(\*SuperscriptBox[\(p\), 
TagBox[
RowBox[{"(", 
RowBox[{"0", ",", "1", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y, z], w[x, y, z] - t0 
\!\(\*SuperscriptBox[\(p\), 
TagBox[
RowBox[{"(", 
RowBox[{"0", ",", "0", ",", "1"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y, z], 
       p[x, y, z]}, {x, y, z} \[Element] mesh, 
      Method -> {"FiniteElement", 
        "InterpolationOrder" -> {u -> 2, v -> 2, w -> 2, 
          p -> 2}}];, {i, 1, nn}]; 

Visualization of error on every 5 steps on the line $x=1/2, z=1/2$ for $u_n=(UX[n],VY[n],WZ[n])$, and $p=P0[n]$

Table[{k t0 // N, 
  Plot[Evaluate[{U[x, y, z, k t0]/UX[k] - 1} /. {x -> 1/2, 
      z -> 1/2}], {y, 0, 1}, PlotRange -> All], 
  Plot[Evaluate[{V[x, y, z, k t0]/VY[k] - 1} /. {x -> 1/2, 
      z -> 1/2}], {y, 0, 1}, PlotRange -> All], 
  Plot[Evaluate[{W[x, y, z, k t0]/WZ[k] - 1} /. {x -> 1/2, 
      z -> 1/2}], {y, 0, 1}, PlotRange -> All], 
  Plot[Evaluate[{P[x, y, z, k t0]/P0[k] - 1} /. {x -> 1/2, 
      z -> 1/2}], {y, 0, 1}, PlotRange -> All]}, {k, 5, nn, 5}]

Please note, that in the picture shown last steps only. The relative error for pressure is about $3\times 10^{-3}$, and for velocity $1.25\times10^{-3}$ on the grid $10\times10\times 10$ on 200 steps in time with time step $\tau =1/400$. Figure 1

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POSTED BY: EDITORIAL BOARD
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