Name: Diffusion in Poiseuille Flow validation
Creator: FlowDeck
License: https://creativecommons.org/licenses/by/4.0/

Validation: 2D Finite Difference, Advection-Diffusion, Laminar flow

Description

Advection-diffusion of a scalar inside a 2D channel with a prescribed parabolic velocity profile, run to steady state. Half of the channel is modelled, with the centreline at y = 0.

The wall is held at scalar value T = 1 and the inlet at T = 0. The centreline and the outlet are zero gradient.

Figure 1: Flow domain: half channel

Simulation Inputs

Input	Value
Material properties	Diffusivity D = 7.143 × 10⁻² m²/s. Péclet number Pe = u₀H/D = 27.65.
Geometry	L = 2.0 m, H = 0.4 m.
Boundary Conditions	Prescribed velocity profile at inlet $u_{x} (y) = u_{0} [1 - {(y / H)}^{2}]$ with centre-line velocity u₀ = 4.937 m/s.

Assumptions and Notes

Flow is steady and laminar, with a fixed velocity field (no momentum solve). Profile is compared at x = 1.0 m (mid-length).

Results

Scalar field in the half channel — Figure 2: Scalar field

Scalar profile at x = 1.0 m vs Graetz solution

Hover or tap the chart to read the values at any point. Click a legend item to show or hide it. FlowDeck values are at cell centres; the lines are completed with the wall boundary values.

Figure 3: Scalar profile at x = 1.0 m vs Graetz solution

Scalar at x = 1.0 m: Graetz solution vs FlowDeck

y (m)	Graetz T	FlowDeck T	Abs. error
0.00125	0.086438	0.083064	0.00337
0.00375	0.086520	0.083143	0.00338
0.00375	0.086520	0.083600	0.00292
0.00625	0.086685	0.083300	0.00339
0.00625	0.086685	0.083757	0.00293
0.00875	0.086932	0.083535	0.00340
0.00875	0.086932	0.083993	0.00294
0.01125	0.087261	0.083849	0.00341
0.01125	0.087261	0.084307	0.00295
0.01375	0.087673	0.084241	0.00343
0.01375	0.087673	0.084700	0.00297
0.01625	0.088167	0.084713	0.00345
0.01625	0.088167	0.085172	0.00300
0.01875	0.088744	0.085262	0.00348
0.01875	0.088744	0.085722	0.00302
0.02125	0.089403	0.085891	0.00351
0.02125	0.089403	0.086352	0.00305
0.02375	0.090144	0.086598	0.00355
0.02375	0.090144	0.087060	0.00308
0.02625	0.090969	0.087385	0.00358
0.02625	0.090969	0.087847	0.00312
0.02875	0.091876	0.088250	0.00363
0.02875	0.091876	0.088714	0.00316
0.03125	0.092865	0.089195	0.00367
0.03125	0.092865	0.089660	0.00321
0.03375	0.093938	0.090219	0.00372
0.03375	0.093938	0.090685	0.00325
0.03625	0.095093	0.091322	0.00377
0.03625	0.095093	0.091790	0.00330
0.03875	0.096332	0.092505	0.00383
0.03875	0.096332	0.092974	0.00336
0.04125	0.097654	0.093768	0.00389
0.04125	0.097654	0.094239	0.00341
0.04375	0.099059	0.095111	0.00395
0.04375	0.099059	0.095584	0.00348
0.04625	0.100547	0.096535	0.00401
0.04625	0.100547	0.097008	0.00354
0.04875	0.102119	0.098038	0.00408
0.04875	0.102119	0.098514	0.00361
0.05125	0.103774	0.099622	0.00415
0.05125	0.103774	0.100099	0.00367
0.05375	0.105513	0.101287	0.00423
0.05375	0.105513	0.101766	0.00375
0.05625	0.107335	0.103033	0.00430
0.05625	0.107335	0.103513	0.00382
0.05875	0.109242	0.104859	0.00438
0.05875	0.109242	0.105342	0.00390
0.06125	0.111232	0.106768	0.00446
0.06125	0.111232	0.107252	0.00398
0.06375	0.113306	0.108757	0.00455
0.06375	0.113306	0.109244	0.00406
0.06625	0.115465	0.110829	0.00464
0.06625	0.115465	0.111318	0.00415
0.06875	0.117707	0.112982	0.00473
0.06875	0.117707	0.113473	0.00423
0.07125	0.120034	0.115217	0.00482
0.07125	0.120034	0.115711	0.00432
0.07375	0.122446	0.117535	0.00491
0.07375	0.122446	0.118031	0.00442
0.07625	0.124942	0.119936	0.00501
0.07625	0.124942	0.120433	0.00451
0.07875	0.127522	0.122419	0.00510
0.07875	0.127522	0.122919	0.00460
0.08125	0.130188	0.124985	0.00520
0.08125	0.130188	0.125487	0.00470
0.08375	0.132938	0.127634	0.00530
0.08375	0.132938	0.128139	0.00480
0.08625	0.135772	0.130367	0.00541
0.08625	0.135772	0.130873	0.00490
0.08875	0.138692	0.133183	0.00551
0.08875	0.138692	0.133692	0.00500
0.09125	0.141696	0.136082	0.00561
0.09125	0.141696	0.136594	0.00510
0.09375	0.144785	0.139066	0.00572
0.09375	0.144785	0.139580	0.00521
0.09625	0.147959	0.142133	0.00583
0.09625	0.147959	0.142649	0.00531
0.09875	0.151218	0.145285	0.00593
0.09875	0.151218	0.145803	0.00541
0.10125	0.154562	0.148521	0.00604
0.10125	0.154562	0.149041	0.00552
0.10375	0.157990	0.151841	0.00615
0.10375	0.157990	0.152364	0.00563
0.10625	0.161504	0.155246	0.00626
0.10625	0.161504	0.155771	0.00573
0.10875	0.165102	0.158735	0.00637
0.10875	0.165102	0.159262	0.00584
0.11125	0.168785	0.162309	0.00648
0.11125	0.168785	0.162838	0.00595
0.11375	0.172552	0.165967	0.00658
0.11375	0.172552	0.166499	0.00605
0.11625	0.176404	0.169711	0.00669
0.11625	0.176404	0.170244	0.00616
0.11875	0.180340	0.173539	0.00680
0.11875	0.180340	0.174074	0.00627
0.12125	0.184360	0.177451	0.00691
0.12125	0.184360	0.177988	0.00637
0.12375	0.188465	0.181449	0.00702
0.12375	0.188465	0.181988	0.00648
0.12625	0.192653	0.185531	0.00712
0.12625	0.192653	0.186072	0.00658
0.12875	0.196925	0.189698	0.00723
0.12875	0.196925	0.190240	0.00668
0.13125	0.201281	0.193949	0.00733
0.13125	0.201281	0.194493	0.00679
0.13375	0.205719	0.198285	0.00743
0.13375	0.205719	0.198830	0.00689
0.13625	0.210241	0.202705	0.00754
0.13625	0.210241	0.203252	0.00699
0.13875	0.214845	0.207209	0.00764
0.13875	0.214845	0.207758	0.00709
0.14125	0.219532	0.211798	0.00773
0.14125	0.219532	0.212347	0.00718
0.14375	0.224300	0.216470	0.00783
0.14375	0.224300	0.217021	0.00728
0.14625	0.229151	0.221226	0.00792
0.14625	0.229151	0.221777	0.00737
0.14875	0.234082	0.226065	0.00802
0.14875	0.234082	0.226617	0.00746
0.15125	0.239095	0.230987	0.00811
0.15125	0.239095	0.231541	0.00755
0.15375	0.244188	0.235992	0.00820
0.15375	0.244188	0.236546	0.00764
0.15625	0.249361	0.241080	0.00828
0.15625	0.249361	0.241635	0.00773
0.15875	0.254613	0.246249	0.00836
0.15875	0.254613	0.246805	0.00781
0.16125	0.259945	0.251501	0.00844
0.16125	0.259945	0.252057	0.00789
0.16375	0.265355	0.256834	0.00852
0.16375	0.265355	0.257390	0.00797
0.16625	0.270843	0.262247	0.00860
0.16625	0.270843	0.262803	0.00804
0.16875	0.276409	0.267742	0.00867
0.16875	0.276409	0.268298	0.00811
0.17125	0.282052	0.273316	0.00874
0.17125	0.282052	0.273872	0.00818
0.17375	0.287770	0.278970	0.00880
0.17375	0.287770	0.279525	0.00825
0.17625	0.293565	0.284702	0.00886
0.17625	0.293565	0.285257	0.00831
0.17875	0.299434	0.290513	0.00892
0.17875	0.299434	0.291067	0.00837
0.18125	0.305377	0.296402	0.00898
0.18125	0.305377	0.296955	0.00842
0.18375	0.311394	0.302368	0.00903
0.18375	0.311394	0.302920	0.00847
0.18625	0.317484	0.308410	0.00907
0.18625	0.317484	0.308962	0.00852
0.18875	0.323646	0.314528	0.00912
0.18875	0.323646	0.315079	0.00857
0.19125	0.329879	0.320722	0.00916
0.19125	0.329879	0.321270	0.00861
0.19375	0.336182	0.326989	0.00919
0.19375	0.336182	0.327537	0.00865
0.19625	0.342556	0.333331	0.00922
0.19625	0.342556	0.333876	0.00868
0.19875	0.348998	0.339745	0.00925
0.19875	0.348998	0.340288	0.00871
0.20125	0.355507	0.346231	0.00928
0.20125	0.355507	0.346772	0.00873
0.20375	0.362084	0.352788	0.00930
0.20375	0.362084	0.353328	0.00876
0.20625	0.368727	0.359416	0.00931
0.20625	0.368727	0.359953	0.00877
0.20875	0.375435	0.366113	0.00932
0.20875	0.375435	0.366647	0.00879
0.21125	0.382208	0.372879	0.00933
0.21125	0.382208	0.373410	0.00880
0.21375	0.389043	0.379712	0.00933
0.21375	0.389043	0.380241	0.00880
0.21625	0.395941	0.386612	0.00933
0.21625	0.395941	0.387138	0.00880
0.21875	0.402900	0.393577	0.00932
0.21875	0.402900	0.394100	0.00880
0.22125	0.409920	0.400607	0.00931
0.22125	0.409920	0.401127	0.00879
0.22375	0.416999	0.407701	0.00930
0.22375	0.416999	0.408217	0.00878
0.22625	0.424136	0.414858	0.00928
0.22625	0.424136	0.415370	0.00877
0.22875	0.431330	0.422076	0.00925
0.22875	0.431330	0.422584	0.00875
0.23125	0.438580	0.429354	0.00923
0.23125	0.438580	0.429859	0.00872
0.23375	0.445885	0.436692	0.00919
0.23375	0.445885	0.437193	0.00869
0.23625	0.453244	0.444089	0.00916
0.23625	0.453244	0.444585	0.00866
0.23875	0.460656	0.451542	0.00911
0.23875	0.460656	0.452034	0.00862
0.24125	0.468120	0.459052	0.00907
0.24125	0.468120	0.459539	0.00858
0.24375	0.475634	0.466616	0.00902
0.24375	0.475634	0.467099	0.00854
0.24625	0.483198	0.474235	0.00896
0.24625	0.483198	0.474712	0.00849
0.24875	0.490810	0.481906	0.00890
0.24875	0.490810	0.482378	0.00843
0.25125	0.498469	0.489628	0.00884
0.25125	0.498469	0.490095	0.00837
0.25375	0.506174	0.497400	0.00877
0.25375	0.506174	0.497862	0.00831
0.25625	0.513924	0.505222	0.00870
0.25625	0.513924	0.505678	0.00825
0.25875	0.521717	0.513091	0.00863
0.25875	0.521717	0.513542	0.00818
0.26125	0.529553	0.521007	0.00855
0.26125	0.529553	0.521452	0.00810
0.26375	0.537431	0.528969	0.00846
0.26375	0.537431	0.529408	0.00802
0.26625	0.545348	0.536974	0.00837
0.26625	0.545348	0.537407	0.00794
0.26875	0.553305	0.545023	0.00828
0.26875	0.553305	0.545450	0.00786
0.27125	0.561300	0.553113	0.00819
0.27125	0.561300	0.553534	0.00777
0.27375	0.569331	0.561244	0.00809
0.27375	0.569331	0.561658	0.00767
0.27625	0.577398	0.569414	0.00798
0.27625	0.577398	0.569822	0.00758
0.27875	0.585500	0.577623	0.00788
0.27875	0.585500	0.578023	0.00748
0.28125	0.593634	0.585868	0.00777
0.28125	0.593634	0.586262	0.00737
0.28375	0.601801	0.594149	0.00765
0.28375	0.601801	0.594536	0.00727
0.28625	0.609999	0.602464	0.00754
0.28625	0.609999	0.602844	0.00715
0.28875	0.618227	0.610812	0.00741
0.28875	0.618227	0.611185	0.00704
0.29125	0.626483	0.619193	0.00729
0.29125	0.626483	0.619559	0.00692
0.29375	0.634767	0.627604	0.00716
0.29375	0.634767	0.627963	0.00680
0.29625	0.643078	0.636046	0.00703
0.29625	0.643078	0.636397	0.00668
0.29875	0.651415	0.644515	0.00690
0.29875	0.651415	0.644859	0.00656
0.30125	0.659775	0.653012	0.00676
0.30125	0.659775	0.653348	0.00643
0.30375	0.668160	0.661536	0.00662
0.30375	0.668160	0.661864	0.00630
0.30625	0.676566	0.670084	0.00648
0.30625	0.676566	0.670405	0.00616
0.30875	0.684994	0.678656	0.00634
0.30875	0.684994	0.678969	0.00602
0.31125	0.693442	0.687251	0.00619
0.31125	0.693442	0.687557	0.00589
0.31375	0.701910	0.695868	0.00604
0.31375	0.701910	0.696166	0.00574
0.31625	0.710395	0.704506	0.00589
0.31625	0.710395	0.704795	0.00560
0.31875	0.718899	0.713164	0.00574
0.31875	0.718899	0.713445	0.00545
0.32125	0.727418	0.721840	0.00558
0.32125	0.727418	0.722113	0.00531
0.32375	0.735954	0.730534	0.00542
0.32375	0.735954	0.730798	0.00516
0.32625	0.744503	0.739244	0.00526
0.32625	0.744503	0.739500	0.00500
0.32875	0.753067	0.747970	0.00510
0.32875	0.753067	0.748218	0.00485
0.33125	0.761643	0.756711	0.00493
0.33125	0.761643	0.756951	0.00469
0.33375	0.770232	0.765465	0.00477
0.33375	0.770232	0.765697	0.00454
0.33625	0.778832	0.774233	0.00460
0.33625	0.778832	0.774456	0.00438
0.33875	0.787442	0.783012	0.00443
0.33875	0.787442	0.783227	0.00422
0.34125	0.796062	0.791803	0.00426
0.34125	0.796062	0.792009	0.00405
0.34375	0.804691	0.800604	0.00409
0.34375	0.804691	0.800802	0.00389
0.34625	0.813328	0.809415	0.00391
0.34625	0.813328	0.809604	0.00372
0.34875	0.821973	0.818235	0.00374
0.34875	0.821973	0.818415	0.00356
0.35125	0.830625	0.827062	0.00356
0.35125	0.830625	0.827234	0.00339
0.35375	0.839282	0.835897	0.00339
0.35375	0.839282	0.836060	0.00322
0.35625	0.847946	0.844739	0.00321
0.35625	0.847946	0.844893	0.00305
0.35875	0.856615	0.853586	0.00303
0.35875	0.856615	0.853732	0.00288
0.36125	0.865288	0.862439	0.00285
0.36125	0.865288	0.862576	0.00271
0.36375	0.873965	0.871297	0.00267
0.36375	0.873965	0.871425	0.00254
0.36625	0.882646	0.880159	0.00249
0.36625	0.882646	0.880278	0.00237
0.36875	0.891330	0.889025	0.00230
0.36875	0.891330	0.889135	0.00219
0.37125	0.900016	0.897894	0.00212
0.37125	0.900016	0.897996	0.00202
0.37375	0.908705	0.906766	0.00194
0.37375	0.908705	0.906859	0.00185
0.37625	0.917396	0.915640	0.00176
0.37625	0.917396	0.915724	0.00167
0.37875	0.926088	0.924516	0.00157
0.37875	0.926088	0.924591	0.00150
0.38125	0.934782	0.933394	0.00139
0.38125	0.934782	0.933460	0.00132
0.38375	0.943476	0.942273	0.00120
0.38375	0.943476	0.942330	0.00115
0.38625	0.952171	0.951152	0.00102
0.38625	0.952171	0.951201	0.00097
0.38875	0.960867	0.960033	0.00083
0.38875	0.960867	0.960073	0.00079
0.39125	0.969563	0.968914	0.00065
0.39125	0.969563	0.968945	0.00062
0.39375	0.978259	0.977796	0.00046
0.39375	0.978259	0.977818	0.00044
0.39625	0.986955	0.986677	0.00028
0.39625	0.986955	0.986691	0.00026
0.39875	0.995652	0.995559	0.00009
0.39875	0.995652	0.995564	0.00009

Reference Data

The reference is the classical Graetz series [1], taking the first 10 eigenfunctions, with coefficients T_m found through integrals of the hypergeometric functions ₁F₁ [2]:

T (x, y) = 1 - \sum_{m = 0}^{10} T_{m} e^{- m^{4} x / (H Pe)} e^{- {(m y / \sqrt{2} H)}^{2}} {}_{1}F_{1} (- \frac{m^{2}}{4} + \frac{1}{4}, \frac{1}{2}, m^{2} \frac{y^{2}}{H^{2}})

References

L. Graetz, “Ueber die Wärmeleitungsfähigkeit von Flüssigkeiten,” Annalen der Physik 254(1), 79–94, 1883. doi:10.1002/andp.18832540106
R. K. Shah and A. L. London, Laminar Flow Forced Convection in Ducts, Advances in Heat Transfer, Supplement 1, Academic Press, New York, 1978.

Data licence. The model results on this page and in the CSV downloads are published by FlowDeck under the Creative Commons Attribution 4.0 licence (CC BY 4.0). The reference values are from the sources listed under References.