Stokes Flow

NekRS provides the option to solve unsteady Stokes flow with constant or variable viscosity. The channel case verifies the Stokes flow solver using the Method of Manufactured Solution (MMS). The problem is solved in a quasi-two-dimensional square domain with edge length 2 and an arbitrary user-specified orientation angle $var element = document.getElementById("moose-equation-e8511469-ff61-4d6c-81a8-e3f7d951f519");katex.render("\\alpha", element, {displayMode:false,throwOnError:false});$ .

Governing equations

The nondimensional unsteady Stokes equations for an incompressible fluid with spatially varying kinematic viscosity are

(1)var element = document.getElementById("moose-equation-43f6c961-c03a-4b2f-98ef-00bd674ee6b1");katex.render("\\begin{aligned} \\frac{\\partial\\boldsymbol{u}}{\\partial t} &= -\\nabla p +\\nabla\\cdot\\left[\\nu(x,y)\\left(\\nabla\\boldsymbol{u}+\\nabla\\boldsymbol{u}^{\\,T}\\right)\\right] +\\boldsymbol{f}, \\\\ \\nabla\\cdot\\boldsymbol{u} &=0, \\end{aligned}", element, {displayMode:true,throwOnError:false});

where $var element = document.getElementById("moose-equation-bd5619cb-22c5-41dd-b729-b825bdbe2ea9");katex.render("\\boldsymbol{u}", element, {displayMode:false,throwOnError:false});$ is the velocity vector, $var element = document.getElementById("moose-equation-84cd8a9c-bc2a-45c0-9821-6c6016b3e4f6");katex.render("p", element, {displayMode:false,throwOnError:false});$ is the hydrodynamic pressure, $var element = document.getElementById("moose-equation-0011b288-7d25-49ec-8fbf-c6a43e09d3d1");katex.render("\\nu", element, {displayMode:false,throwOnError:false});$ is the kinematic viscosity, and $var element = document.getElementById("moose-equation-325a0db6-23c3-4eba-9ba7-06018dc1458a");katex.render("\\boldsymbol{f}", element, {displayMode:false,throwOnError:false});$ is a prescribed forcing function. The nonlinear advection term is omitted from the momentum equation, distinguishing Stokes flow from the full incompressible Navier-Stokes equations.

Manufactured solution

The manufactured steady-state velocity solution is

(2)var element = document.getElementById("moose-equation-96e3a9bb-7df6-44f0-9a79-390842918cca");katex.render("\\begin{aligned} u' &= U_0\\cos\\left[\\pi(x+0.5)\\right]\\sin\\left[\\pi(2y+0.5)\\right], \\\\ v' &= -\\frac{U_0}{2}\\sin\\left[\\pi(x+0.5)\\right]\\cos\\left[\\pi(2y+0.5)\\right], \\\\ u(x,y) &= u'\\cos(\\alpha)-v'\\sin(\\alpha), \\\\ v(x,y) &= u'\\sin(\\alpha)+v'\\cos(\\alpha). \\end{aligned}", element, {displayMode:true,throwOnError:false});

The manufactured viscosity field is

(3)var element = document.getElementById("moose-equation-3bdd339e-85fc-4885-b75b-98897d914f38");katex.render("\\nu(x,y)=\\frac{1}{Re}(1+ay),", element, {displayMode:true,throwOnError:false});

where $var element = document.getElementById("moose-equation-69011779-ce3e-4be5-834f-9710476caf12");katex.render("Re", element, {displayMode:false,throwOnError:false});$ is the Reynolds number, $var element = document.getElementById("moose-equation-a9800450-ce15-4b33-9451-6adf9a49fabb");katex.render("U_0", element, {displayMode:false,throwOnError:false});$ is a user-specified velocity scale, $var element = document.getElementById("moose-equation-94f0daed-bd29-4f7b-94d3-5c4c1941167f");katex.render("a", element, {displayMode:false,throwOnError:false});$ is a user-specified viscosity scaling parameter, and $var element = document.getElementById("moose-equation-3b27760a-b090-4075-97e2-e366a5136a5b");katex.render("u", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-e2676301-0369-4b5a-bc92-ba3f05734689");katex.render("v", element, {displayMode:false,throwOnError:false});$ are the velocity components in the $var element = document.getElementById("moose-equation-5d0c3aea-bfc8-42cd-a9db-1c42cc56b75a");katex.render("x", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-573360c1-10e3-4f40-9149-c130e2434822");katex.render("y", element, {displayMode:false,throwOnError:false});$ directions, respectively. The spatial variation in viscosity introduces an additional contribution to the viscous operator through $var element = document.getElementById("moose-equation-45823b14-4707-4a39-a284-d38fa0684def");katex.render("\\nabla\\nu", element, {displayMode:false,throwOnError:false});$ . The corresponding manufactured forcing function is

(4)var element = document.getElementById("moose-equation-169fd4e9-d6eb-4d13-8870-1793129dcb0f");katex.render("\\boldsymbol{f} = -\\nabla\\nu\\cdot \\left(\\nabla\\boldsymbol{u}+\\nabla\\boldsymbol{u}^{\\,T}\\right).", element, {displayMode:true,throwOnError:false});

Verification criteria

The simulations are performed using a polynomial order of seven and two geometrical configurations. The first configuration uses the original geometry, while the second rotates the geometry by $var element = document.getElementById("moose-equation-20c59474-4676-4108-b016-984949033e65");katex.render("45^\\circ", element, {displayMode:false,throwOnError:false});$ . Errors are evaluated at $var element = document.getElementById("moose-equation-f9f8a9f4-be00-4ca5-bc37-6d37a4c7dd78");katex.render("t=0.1", element, {displayMode:false,throwOnError:false});$ ; because the initial condition is the exact solution, this time is short enough for the test to run quickly, but long enough such that any code changes or inconsistencies in the implementation would manifest as errors. The volume-integrated velocity error is defined as

(5)var element = document.getElementById("moose-equation-8123dd94-ca51-43f6-afd3-66e69f3961ab");katex.render("\\varepsilon_{\\boldsymbol{u}} = \\left\\lVert\\boldsymbol{u}-\\boldsymbol{u}_{\\mathrm{exact}}\\right\\rVert_{L_2(\\Omega)} = \\left[ \\int_\\Omega \\left|\\boldsymbol{u}-\\boldsymbol{u}_{\\mathrm{exact}}\\right|^2 \\,d\\Omega \\right]^{1/2}.", element, {displayMode:true,throwOnError:false});

Figure 1 and Figure 2 present the volume-integrated error norms for the two configurations. The results demonstrate spectral convergence of the velocity solution, thereby verifying the accuracy of the Stokes flow solver.

Volume-integrated velocity error norms for the Stokes flow case using original geometry

Figure 1: Volume-integrated error norms for the channel case with original geometry

Volume-integrated velocity error norms for the Stokes flow case using rotated geometry

Figure 2: Volume-integrated error norms for the channel case using rotated geometry

Governing equations
Manufactured solution
Verification criteria