This test case solves the Poiseuille flow with immersed boundary conditions.

This test case suite focuses on the second-order immersed boundary methods applied to the incompressible Navier-Stokes equation with Dirichlet boundary condition. The objectives of this test case are:

verify the core of the immersed boundary methods.
verify the order of spatial convergence of the methods.

For more details, see [1] and [2].

Configurations

Physical domain.

The global boundaries are defined by the intevales:

\begin{align} [0, 2] &\quad\text{in the flow direction,} & [-1, +1] &\quad\text{otherwise.} \end{align}

The immersed boundary is either a flat channel or a cylinder (not treated so far):

\begin{align} y = \pm h &\quad\text{(flatchannel),} & x^2 + y^2 = h^2 &\quad\text{(cylinder),} \end{align}

where \(h = \frac{1}{2} \frac{262144}{177147} \), which is half the wolf interval of the Pythagorean scale. This has been chosed to avoid the boundary to match any grid.

Solutions

The velocity has the following parabolic shape: solution:

\begin{align} u(y) = \hat{u} \left(1 - \frac{y^2}{h^2}\right) &\quad\text{(flatchannel),} & u(r) = \hat{u} \left(1 - \frac{r^2}{h^2}\right) &\quad\text{(cylinder).} \end{align}

The flow is pushed by the navier source term, which should be equal to:

\begin{align} f_x = 2 \frac{\mu\hat{u}}{h^2} &\quad\text{(flatchannel),} & f_y = 4 \frac{\mu\hat{u}}{h^2} &\quad\text{(cylinder),} \end{align}

where \( \mu \) is the dynamic viscosity of the fluid. Using the navier source term yields a constant pressure gradient, which allows the domain to be periodic in the flow direction.

The average velocity is computed from the integral:

\begin{align} V \bar{u} = \int_{-h}^{+h}{u(y)\mathrm{d}y} = 2 h \hat{u} \int_0^1 {(1 - z^2) \mathrm{d}z} &\quad\text{(flatchannel),} \\ V \bar{u} = \int_{0}^{2\pi}\int_{0}^{+h}{u(r)r\mathrm{d}r\mathrm{d}\theta} = 2 \pi h^2 \hat{u} \int_0^1 {(1 - z^2) z \mathrm{d}z} &\quad\text{(cylinder),} \end{align}

giving:

\begin{align} \bar{u} = \frac{2}{3}\hat{u} &\quad\text{(flatchannel),} & \bar{u} = \frac{1}{2}\hat{u} &\quad\text{(cylinder).} \end{align}

In the actual test case, the dynamic viscosity of the fluid is set to match a given Reynolds number, which is defined by:

\begin{align} \mathit{Re} = \frac {2h \times \bar{u}} {\mu} \end{align}

We choose to set \(\bar{u}\) and \(\mu\) as follows:

\begin{align} \bar{u} &= \frac {1} {2h} & \mu = \frac {1} {\mathit{Re}} \end{align}

Values are given in the table below.

Geometry	\( \bar{u} \)
flatchannel	\( \tfrac{59049}{131072} = 0.45050811767578125\ldots \)
cylinder	\( \tfrac{177147}{524288} = 0.33788108825683594\ldots \)

and \( \mathit{Re} = 20\).

Runtime parameters

The following common settings are used:

Second ordered centered scheme are used
Several immersed boundary methods are verified on this test case: direct, linear, LGS, LIS, Q, QO, QGS, QGSO, QIS, QISS1 (see [2] for details).

Results

2D Poiseuille flow

Velocity errors and pressure \(L^2\) error show a second-order limiting behavior, while pressure \(L_\infty\) error show a limiting behavior between 1.5 and 2.

Direct

Mesh	Velocity L1 error	order	Velocity L2 error	order	Velocity Linf error	order	Pressure L1 error	order	Pressure L2 error	order	Pressure Linf error	order
20×20	0.05270009229508642	n/a	0.029774892943298048	n/a	0.02979918828191397	n/a	0.08371826403250907	n/a	0.06227244620088994	n/a	0.11982979072503541	n/a
40×40	0.012411033985466223	2.086	0.006948353359337286	2.099	0.007728497257455225	1.947	0.020386872969755852	2.038	0.014799064758676709	2.073	0.03138706695860538	1.933
80×80	0.00323814713572379	1.938	0.001802675947358788	1.947	0.0019286508605681485	2.003	0.005966030307143693	1.773	0.004283405365244087	1.789	0.012300211461270472	1.351
160×160	0.0008023379002303047	2.013	0.00044783136332388247	2.009	0.0004619905365337573	2.062	0.0015015093832701394	1.990	0.0010986920794422727	1.963	0.005611819837253673	1.132

Linear

Mesh	Velocity L1 error	order	Velocity L2 error	order	Velocity Linf error	order	Pressure L1 error	order	Pressure L2 error	order	Pressure Linf error	order
20×20	0.06482457588961568	n/a	0.037356577161486965	n/a	0.06731343557486727	n/a	0.07741552084488161	n/a	0.06980344871362482	n/a	0.12542430184538222	n/a
40×40	0.016613319351544923	1.964	0.00985937260684884	1.922	0.017408597526763627	1.951	0.02237926524669858	1.790	0.020127331576449116	1.794	0.050675474706367574	1.307
80×80	0.004504902911414919	1.883	0.0025315385727960366	1.961	0.0043857053639653806	1.989	0.010874739800558027	1.041	0.007808536419449842	1.366	0.025348211598308357	0.999
160×160	0.0010605643096097718	2.087	0.0005995106253329054	2.078	0.001097395324387768	1.999	0.002680683517235237	2.020	0.00194527373415673	2.005	0.005208202868149936	2.283

LGS

Mesh	Velocity L1 error	order	Velocity L2 error	order	Velocity Linf error	order	Pressure L1 error	order	Pressure L2 error	order	Pressure Linf error	order
20×20	0.05382527244907745	n/a	0.030287990652740692	n/a	0.02997894922617972	n/a	0.08683306127071988	n/a	0.06439819519322483	n/a	0.12186089843871883	n/a
40×40	0.012634056886718373	2.091	0.007074169680670298	2.098	0.007916064972680992	1.921	0.021587302908959972	2.008	0.015638386795382694	2.042	0.032917627248945536	1.888
80×80	0.0032711559672135033	1.949	0.0018215547868255166	1.957	0.0019647604528806872	2.010	0.006169418510733906	1.807	0.004430453092171552	1.820	0.012983587662725338	1.342
160×160	0.0008096416484076865	2.014	0.000452366651513424	2.010	0.00047507345904762577	2.048	0.0015493197346734726	1.993	0.0011322595899753971	1.968	0.005579876438810416	1.218

LIS

Mesh	Velocity L1 error	order	Velocity L2 error	order	Velocity Linf error	order	Pressure L1 error	order	Pressure L2 error	order	Pressure Linf error	order
20×20	0.05385389606746297	n/a	0.030319936831729823	n/a	0.029733939786157126	n/a	0.08676312210840181	n/a	0.06429186006954875	n/a	0.12138424913379575	n/a
40×40	0.012748740567615557	2.079	0.007141147800969571	2.086	0.00806035778721248	1.883	0.021973314021904153	1.981	0.015928598383612474	2.013	0.03556822694945705	1.771
80×80	0.0033006903055519876	1.950	0.0018386693313723341	1.957	0.002003110021105481	2.009	0.006320981573926354	1.798	0.004537169738851249	1.812	0.014247415843120947	1.320
160×160	0.0008121841359176234	2.023	0.00045468697879093	2.016	0.0004872933032785687	2.039	0.0015648387424484504	2.014	0.0011391219683095676	1.994	0.004440394342886567	1.682

Q

Mesh	Velocity L1 error	order	Velocity L2 error	order	Velocity Linf error	order	Pressure L1 error	order	Pressure L2 error	order	Pressure Linf error	order
20×20	0.05224468511521704	n/a	0.029958970737931204	n/a	0.049816724958424596	n/a	0.07156071476218151	n/a	0.05698485159760314	n/a	0.12035411577764554	n/a
40×40	0.013109446966161061	1.995	0.007276207393553354	2.042	0.012082183161878236	2.044	0.012179075523347333	2.555	0.012174175782841471	2.227	0.0351372017790057	1.776
80×80	0.003837420189889581	1.772	0.0021817806559658744	1.738	0.003019957018883559	2.000	0.007195966258691036	0.759	0.005422724916079778	1.167	0.01770721828104982	0.989

QGS

Mesh	Velocity L1 error	order	Velocity L2 error	order	Velocity Linf error	order	Pressure L1 error	order	Pressure L2 error	order	Pressure Linf error	order
20×20	0.048682849684619035	n/a	0.02767816161710693	n/a	0.032485609938977644	n/a	0.05663355057434309	n/a	0.04572409373515861	n/a	0.10977011394925773	n/a
40×40	0.011202189264120636	2.120	0.006193874593197813	2.160	0.006699085682877581	2.278	0.009430112586382558	2.586	0.007667532246180057	2.576	0.020809103713967048	2.399
80×80	0.0031566420455928093	1.827	0.0017590782669031548	1.816	0.0019650731891998063	1.769	0.004388165601259698	1.104	0.003211606701204337	1.255	0.009394162190362376	1.147
160×160	0.0007577379845984029	2.059	0.0004200677564264471	2.066	0.0004373645634714318	2.168	0.0010660787018279896	2.041	0.0008157507422324713	1.977	0.0030584649899583205	1.619

QIS

Mesh	Velocity L1 error	order	Velocity L2 error	order	Velocity Linf error	order	Pressure L1 error	order	Pressure L2 error	order	Pressure Linf error	order
20×20	0.05011754592367867	n/a	0.02863006219680653	n/a	0.03367981066941761	n/a	0.056019447314914296	n/a	0.04678053809729675	n/a	0.11263389597080531	n/a
40×40	0.01117332096589729	2.165	0.00613941236288884	2.221	0.006737017557861047	2.322	0.00906304590034595	2.628	0.007350896890872159	2.670	0.0201544662561588	2.482
80×80	0.003110939266274319	1.845	0.001722607406276905	1.834	0.0019371216308767814	1.798	0.0038343739223206342	1.241	0.0028334003134005714	1.375	0.008070710141589998	1.320
160×160	0.0007406022133274162	2.071	0.000409352038212465	2.073	0.0004273355314918792	2.180	0.0009332289453220675	2.039	0.0007352277981699719	1.946	0.0025862628593291692	1.642

QISS1

Mesh	Velocity L1 error	order	Velocity L2 error	order	Velocity Linf error	order	Pressure L1 error	order	Pressure L2 error	order	Pressure Linf error	order
20×20	0.05270009229508629	n/a	0.029774892943297875	n/a	0.029799188281913747	n/a	0.08371826403254023	n/a	0.062272446200916265	n/a	0.11982979072508781	n/a
40×40	0.012411033985466188	2.086	0.0069483533593372995	2.099	0.007728497257455336	1.947	0.020386872968579293	2.038	0.01479906475789101	2.073	0.03138706695765281	1.933
80×80	0.003238147135723599	1.938	0.0018026759473586881	1.947	0.0019286508605681485	2.003	0.005966030307135308	1.773	0.004283405365238609	1.789	0.012300211461274468	1.351
160×160	0.0008023378973876549	2.013	0.0004478313579448087	2.009	0.0004619905261575852	2.062	0.0015015514384990962	1.990	0.0010987242150504754	1.963	0.005611885903512359	1.132

2D Poiseuille flow (cylinder)

Second-order convergence is obtained for pressure with the L2 error norm. It should be noted that the ﬁrst-order convergence for the \(L∞\) norm of pressure is expected as it is also encountered for the canonical validation case of 2D channel Poiseuille ﬂow without immersed boundaries. Also as expected, the LIS and QIS methods are more accurate than the L method and the L∞ error norm of pressure has a slightly better order of convergence.

As the analytical solution is parabolic, the used of QISS1 with third-order Lagrange interpolations (p=3) yields error ﬁelds with values close to machine epsilon whilst maintaining a stencil size of 2 as the L method.

Linear

Mesh	Velocity L1 error	order	Velocity L2 error	order	Velocity Linf error	order	Pressure L1 error	order	Pressure L2 error	order	Pressure Linf error	order
16×16×16	0.010337386766712848	n/a	0.005657007393059307	n/a	0.006428902652152874	n/a	0.008805495521884449	n/a	0.006020993385041923	n/a	0.01108876484035437	n/a
32×32×32	0.0024916611945808693	2.053	0.0014352360432396119	1.979	0.002546293554459112	1.336	0.0021833984031973197	2.012	0.0015566628515310558	1.952	0.006218550865204342	0.834
64×64×64	0.0006931981221093784	1.846	0.00039042182068771246	1.878	0.0007957172525748868	1.678	0.0006180878339082197	1.821	0.0004579762142514126	1.765	0.003516024618832575	0.823

LIS

Mesh	Velocity L1 error	order	Velocity L2 error	order	Velocity Linf error	order	Pressure L1 error	order	Pressure L2 error	order	Pressure Linf error	order
16×16×16	0.008676246062052978	n/a	0.004734634567195655	n/a	0.006114451773218135	n/a	0.007390249953530459	n/a	0.005072826416071411	n/a	0.010224860446700429	n/a
32×32×32	0.0018911785004843728	2.198	0.0010543271804403505	2.167	0.001798528830088672	1.765	0.0016667851078954422	2.149	0.0011900775044682427	2.092	0.004882407614013928	1.066
64×64×64	0.0004952284961289842	1.933	0.0002666318422913067	1.983	0.0004614460158981386	1.963	0.0004425526793380094	1.913	0.0003263767585142331	1.866	0.002441597814446611	1.000

QIS

Mesh	Velocity L1 error	order	Velocity L2 error	order	Velocity Linf error	order	Pressure L1 error	order	Pressure L2 error	order	Pressure Linf error	order
16×16×16	0.004903861652830974	n/a	0.002813885379504828	n/a	0.004503557234462954	n/a	0.003995228007342633	n/a	0.0027815397710551147	n/a	0.00685432195525415	n/a
32×32×32	0.001109368096433153	2.144	0.0006256407600960024	2.169	0.001082908522228962	2.056	0.0009582823486652074	2.060	0.000685726791334464	2.020	0.0032873392118683564	1.060
64×64×64	0.00029851197389575804	1.894	0.00016128648005764277	1.956	0.0002789810518378814	1.957	0.0002617345438629307	1.872	0.00019357443284158615	1.825	0.0016487843625357657	0.996

QISS1P3

Mesh	Velocity L1 error	order	Velocity L2 error	order	Velocity Linf error	order	Pressure L1 error	order	Pressure L2 error	order	Pressure Linf error	order
16×16×16	4.183202951107421e-11	n/a	2.7435545131984187e-11	n/a	7.872003049413934e-11	n/a	1.77051719172689e-11	n/a	1.1964099720613176e-11	n/a	2.754091399381764e-11	n/a
32×32×32	2.6100136619687503e-11	0.681	1.5894194323673115e-11	0.788	4.7954968919335084e-11	0.715	8.369435737565587e-11	-2.241	4.620983945424045e-11	-1.949	8.486936153850877e-11	-1.624
64×64×64	2.2466659715245315e-11	0.216	1.542234976989625e-11	0.043	4.3143058943698217e-10	-3.169	2.8362573223258807e-10	-1.761	1.5312400022022985e-10	-1.728	1.2615800626392115e-09	-3.894

Reference

[1] J. Picot, S. Glockner, Discretization stencil reduction of direct forcing immersed boundary methods on rectangular cells: the Ghost Node Shifting Method, Journal of Computational Physics, 364, pp18-48, 2018.

[2] A. M. D. Jost and S. Glockner, Direct forcing immersed boundary methods: Improvements to the Ghost Node Method, Journal of Computational Physics, volume 438, 110371, 2021.