极低雷诺数下，PIMPLE迭代初始残差保持为1，降不下来

lwjetmann

请教各位！！李老师说，扩散项增加对角占优，有助于收敛。那如果雷诺数给极低，理论上会出现什么问题吗？

我使用pimpleFoam+动网格计算一竖排圆柱（直径为d，间距为10d）从静止开始加速平动，雷诺数给1e-3（基于d）。5个圆柱及以下，计算正常；9个圆柱及以上，计算出现问题（增大雷诺数至1e-2则可正常计算）。相邻圆柱的网格是对称的，增加圆柱个数其实单纯增加网格量而已。

症状：

圆柱数量多的情况，每次PIMPLE迭代（nOuterCorrectors）的压力方程初始残差降不下来也不发散

Courant Number mean: 2.398253992e-06 max: 0.0215123576
deltaT = 0.04
Time = 6.7

PIMPLE: iteration 1
DICPCG:  Solving for pcorr, Initial residual = 1, Final residual = 0.09682197649, No Iterations 82
DICPCG:  Solving for pcorr, Initial residual = 0.09688756857, Final residual = 0.00927208952, No Iterations 82
DICPCG:  Solving for pcorr, Initial residual = 0.01741032791, Final residual = 0.00171820424, No Iterations 86
DICPCG:  Solving for pcorr, Initial residual = 0.002315321063, Final residual = 0.0002293036835, No Iterations 341
DICPCG:  Solving for pcorr, Initial residual = 0.0005365215345, Final residual = 9.975987647e-10, No Iterations 791
time step continuity errors : sum local = 1.735383521e-22, global = 6.725543895e-25, cumulative = 3.656703828e-10
GAMG:  Solving for p, Initial residual = 0.9443291304, Final residual = 0.002194485674, No Iterations 3
GAMG:  Solving for p, Initial residual = 0.001947671658, Final residual = 1.846802379e-05, No Iterations 35
GAMG:  Solving for p, Initial residual = 7.588154638e-05, Final residual = 7.535974383e-07, No Iterations 40
GAMG:  Solving for p, Initial residual = 4.120836164e-06, Final residual = 4.025364261e-08, No Iterations 50
GAMG:  Solving for p, Initial residual = 7.005626725e-07, Final residual = 6.534878288e-09, No Iterations 45
time step continuity errors : sum local = 6.700624174e-17, global = 1.044778689e-17, cumulative = 3.656703932e-10
GAMG:  Solving for p, Initial residual = 0.955093981, Final residual = 0.002181673685, No Iterations 3
GAMG:  Solving for p, Initial residual = 0.001764171308, Final residual = 1.648106932e-05, No Iterations 35
GAMG:  Solving for p, Initial residual = 7.35621349e-05, Final residual = 7.223433176e-07, No Iterations 42
GAMG:  Solving for p, Initial residual = 3.059263044e-06, Final residual = 3.023418303e-08, No Iterations 62
GAMG:  Solving for p, Initial residual = 3.427802208e-07, Final residual = 9.5003603e-10, No Iterations 84
time step continuity errors : sum local = 9.859629733e-18, global = -1.244226379e-18, cumulative = 3.65670392e-10
PIMPLE: iteration 2
...
PIMPLE: iteration 29
GAMG:  Solving for p, Initial residual = 0.9411617368, Final residual = 0.00215235933, No Iterations 3
GAMG:  Solving for p, Initial residual = 0.001918610458, Final residual = 1.812341162e-05, No Iterations 35
GAMG:  Solving for p, Initial residual = 7.116434841e-05, Final residual = 6.94389095e-07, No Iterations 43
GAMG:  Solving for p, Initial residual = 3.125063719e-06, Final residual = 3.121354786e-08, No Iterations 60
GAMG:  Solving for p, Initial residual = 3.47115616e-07, Final residual = 3.269278386e-09, No Iterations 53
time step continuity errors : sum local = 3.261140983e-17, global = 4.05973866e-18, cumulative = 3.656704773e-10
GAMG:  Solving for p, Initial residual = 0.9520350913, Final residual = 0.00217738456, No Iterations 3
GAMG:  Solving for p, Initial residual = 0.001759034352, Final residual = 1.660230344e-05, No Iterations 35
GAMG:  Solving for p, Initial residual = 7.423394075e-05, Final residual = 7.303987078e-07, No Iterations 42
GAMG:  Solving for p, Initial residual = 3.089411582e-06, Final residual = 3.055292623e-08, No Iterations 62
GAMG:  Solving for p, Initial residual = 3.343358255e-07, Final residual = 9.575916541e-10, No Iterations 84
time step continuity errors : sum local = 9.662150084e-18, global = -1.230822196e-18, cumulative = 3.65670476e-10

n21_re0.001.png

流场明显发散了

分析：

试过修改离散格式，问题可能不在于此。时间项改用backward不影响；对流项采用linear或linearUpwind没有区别（毕竟对流项太弱）；扩散项用的corrected；梯度项用linear或leastSquares（或加上限制器）都不影响其发散。

ddtSchemes
{
    default         Euler;
}

gradSchemes
{
    default         cellMDLimited leastSquares 0.5;
}

divSchemes
{
    default         none;
    div(phi,U)      bounded Gauss linearUpwind grad(U);
    div((nuEff*dev2(T(grad(U))))) Gauss linear;
}

laplacianSchemes
{
    default         Gauss linear corrected;
}

试过修改计算域及边界条件，问题可能不在于此。计算域给10000d或更大，至少在初始运动阶段，该域是足够大的，但是都会很快发散。边界条件可动的地方也不多，把壁面p从零梯度改为fixedFluxPressure没用

# U
    "cylinder.*"
    {
        type            movingWallVelocity;
        value           uniform (0 0 0);
    }

    "(up|bottom|inlet|outlet)"
    {
        type            pressureInletOutletVelocity;
        value                   uniform (0 0 0);
    }

# p
    "cylinder.*"
    {
        type            fixedFluxPressure;
        //type            zeroGradient;
        value			uniform 0;
    }
    
    "(up|bottom|inlet|outlet)"
    {
        type            outletInlet;
        outletValue 	 uniform 0;
    }

时间步长，迭代次数精度等，都修改过，也不影响其最终发散

solvers
{
    p
    {
        solver          GAMG;
        smoother        GaussSeidel;
        tolerance       1e-09;
        relTol          0.1;
    }
    pFinal
    {
        solver          GAMG;
        smoother        GaussSeidel;
        tolerance       1e-09;
        relTol          0;
    }
    pcorr
    {
        solver          PCG;
        smoother        GaussSeidel;
        tolerance       1e-09;
        relTol          0.1;
        preconditioner  DIC;
    }
    pcorrFinal
    {
        solver          PCG;
        smoother        GaussSeidel;
        tolerance       1e-09;
        relTol          0;
        preconditioner  DIC;
    }
}

PIMPLE
{
    momentumPredictor off;
    correctPhi      yes;
    nOuterCorrectors 500;
    nCorrectors     2;
    nNonOrthogonalCorrectors 4;
    ddtCorr         true;
    residualControl
    {
        p
        {
            tolerance       1e-08;
            relTol          0;
        }
    }
}

relaxationFactors
{
    equations
    {
        ".*"            1;
    }
}

网格。从流场上看，跟网格关系很大。但按理说低雷诺数计算对网格并不敏感，而且网格质量不差（平均non-orthogonality只6.5），网格非正交修正也都收敛了。雷诺数高一点能算，少几个圆柱也能算。

这里面有没有隐含着什么低雷诺数计算的理论缺陷呢？

李东岳

把你的动量预测关掉看一下。

我最近在美国，更深入的讨论得回法国之后的。

lwjetmann

@李东岳感谢老师回复！动量预测是关掉了的。

打开试了试，步长极小，残差也一直保持在一个值不下降，cumulative则很大，如下：

Courant Number mean: 7.953414508e-05 max: 0.4993036972
deltaT = 4.979675437e-35
Time = 6.00001e-05

PIMPLE: iteration 1
DICPCG:  Solving for pcorr, Initial residual = 1, Final residual = 0.04941709914, No Iterations 29
DICPCG:  Solving for pcorr, Initial residual = 0.03122316034, Final residual = 0.001556737496, No Iterations 121
DICPCG:  Solving for pcorr, Initial residual = 0.006804280077, Final residual = 0.0003314734696, No Iterations 359
DICPCG:  Solving for pcorr, Initial residual = 0.002308369425, Final residual = 9.709852492e-11, No Iterations 911
time step continuity errors : sum local = 1.793718362e-20, global = 1.178518414e-23, cumulative = 1.037419916e+19
smoothSolver:  Solving for Ux, Initial residual = 3.985498772e-05, Final residual = 2.227155701e-11, No Iterations 1
smoothSolver:  Solving for Uy, Initial residual = 6.302253167e-05, Final residual = 2.385619185e-11, No Iterations 1
GAMG:  Solving for p, Initial residual = 0.004038713667, Final residual = 0.0001722133227, No Iterations 18
GAMG:  Solving for p, Initial residual = 0.0005583071565, Final residual = 2.673668049e-05, No Iterations 34
GAMG:  Solving for p, Initial residual = 9.485975434e-05, Final residual = 4.53317788e-06, No Iterations 33
GAMG:  Solving for p, Initial residual = 2.596708738e-05, Final residual = 1.231466526e-06, No Iterations 38
time step continuity errors : sum local = 7.929103562e-15, global = -8.302368146e-16, cumulative = 1.037419916e+19
GAMG:  Solving for p, Initial residual = 0.003717218882, Final residual = 0.000164557501, No Iterations 17
GAMG:  Solving for p, Initial residual = 0.0005378364039, Final residual = 2.53035408e-05, No Iterations 34
GAMG:  Solving for p, Initial residual = 9.265726415e-05, Final residual = 4.535601234e-06, No Iterations 36
GAMG:  Solving for p, Initial residual = 2.556961887e-05, Final residual = 9.918214454e-11, No Iterations 548
time step continuity errors : sum local = 6.482264028e-19, global = 6.603865839e-20, cumulative = 1.037419916e+19
PIMPLE: iteration 2
...
PIMPLE: iteration 50
smoothSolver:  Solving for Ux, Initial residual = 1.226945138e-10, Final residual = 6.6691269e-16, No Iterations 1
smoothSolver:  Solving for Uy, Initial residual = 1.651845516e-10, Final residual = 3.177048458e-16, No Iterations 1
GAMG:  Solving for p, Initial residual = 0.000949721577, Final residual = 4.516299707e-05, No Iterations 41
GAMG:  Solving for p, Initial residual = 0.0004457930337, Final residual = 2.206483819e-05, No Iterations 23
GAMG:  Solving for p, Initial residual = 7.977007558e-05, Final residual = 3.747481321e-06, No Iterations 36
GAMG:  Solving for p, Initial residual = 2.418915618e-05, Final residual = 1.162060008e-06, No Iterations 48
time step continuity errors : sum local = 7.482194647e-15, global = -7.221446192e-16, cumulative = 1.037419916e+19
GAMG:  Solving for p, Initial residual = 0.003705959955, Final residual = 0.0001646823634, No Iterations 17
GAMG:  Solving for p, Initial residual = 0.0005377574684, Final residual = 2.535476181e-05, No Iterations 34
GAMG:  Solving for p, Initial residual = 9.241219827e-05, Final residual = 4.525895839e-06, No Iterations 36
GAMG:  Solving for p, Initial residual = 2.542969083e-05, Final residual = 9.774400044e-11, No Iterations 548
time step continuity errors : sum local = 6.388632178e-19, global = 6.502735317e-20, cumulative = 1.037419916e+19
PIMPLE: not converged within 50 iterations
ExecutionTime = 21923.2 s  ClockTime = 21963 s

Courant Number mean: 7.964991508e-05 max: 0.4989139005
deltaT = 4.99051583e-35
Time = 6.00001e-05

看到站内其它的帖子，试了试有量纲/无量纲计算（此前圆柱直径0.05，nu=50；改为圆柱直径0.0005，nu=0.5），也无法收敛。

走投无路试了一下StarCCM，多低雷诺数都能算，和OpenFOAM那些能收敛的算例比得还贼好

CFD中文网

极低雷诺数下，PIMPLE迭代初始残差保持为1，降不下来