Linear acoustic pulse propagation.
This test case investigates the isothermal problem of a linear acoustic wave propagation considering an inviscid perfect gas fluid ( \mu=0) with its equation of state
\Delta p = c_0^2 \Delta \rho,
with \Delta p the pressure perturbation, \Delta \rho the density perturbation and c_0=\sqrt{\gamma R T_0} the constant speed of sound of the medium. This benchmark has been used in the past to test several novel compressible solvers [1,2,3,4,5,6] often to carry out temporal convergence studies. Besides its simplicity and the existence of analytical solutions, this case allows a clear evaluation of the numerical diffusion and dispersion of the proposed numerical schemes.
We consider a monodimensional periodic domain of length L=1m. For velocity boundary conditions, left and right boundaries have periodic conditions while top and bottom have slip conditions. At initial time, we consider the thermodynamic state (T_0,p_0,\rho_0)=(300K,10^5Pa, \frac{p_0}{R T_0}) and a Gaussian acoustic pressure wave defined as
p(x,t_0) = p_0 + \Delta p_0 \exp(-\frac{x^2}{2\Sigma^2}),
with \Delta p_0 the pulse amplitude and \Sigma a pulse length control parameter. The initial parameter of the pulse is set to \Delta p_0=10^2Pa and \Sigma=0.1m like in [5]. The dimensionless parameters of the problem are respectively \mathrm{Re}_0 = \infty and \mathrm{Ma}_0 = 7.14 \ 10^{4}.
From the resolution of the d'Alembert equation, analytical solutions are available for all fields. The pressure, density and velocity solutions are respectively
p(x,t) = p_0 + \Delta p_0 \exp\left(-\frac{\left(x-c_0 t\right)^2}{2\Sigma^2}\right)\, ,
\rho(x,t) = \rho_0 + \frac{\Delta p_0}{c_0^2} \exp\left(-\frac{\left(x-c_0 t\right)^2}{2\Sigma^2}\right),
u(x,t) = \frac{\Delta p_0}{\rho_0 c_0^2} \exp\left(-\frac{\left(x-c_0 t\right)^2}{2\Sigma^2}\right),
with c_0 t the distance travelled by the wave.
Main parameters to run this test case are:
Next figure (a) presents a graphical temporal convergence study of the relative pressure field at t_f=c_0/L=2.88 \ 10{-3}s (time travelled by the wave until it returns to its initial position) for various acoustic Courant number, noted CFL. The implicit treatment of pressure increment avoids a stability limitation related to acoustic time step as we do not find any stability limit (still stable at CFL=4000, data not shown). For very large CFL and Euler backward temporal scheme, the acoustic wave is totally diffused but, note that for CFL=4,8 the wave is still well predicted. We observe the relative low diffusivity of the first-order temporal scheme Euler backward at CFL=4 compared to literature results [5] which obtain similar value of the maximum of the relative pressure with a low Courant number value (see CFL=0.5 numerized curve from [5]).
Additionally, it is noteworthy, on part (b) of the figure, that the BDF2 scheme, with second-order temporal accuracy, exhibits significantly lower numerical diffusion compared to the Euler scheme. This results in a pressure profile that closely aligns with the exact solution at CFL=2. An error of less than 1\% is observed compared to 20\% with the Euler scheme. Using the BDF2 temporal scheme, the correct observation of acoustic propagation is possible while considering CFLs greater than unity.
In next tables, the temporal convergence study of this test case with the BDF2 scheme is presented with a final time t_f=L/c_s=2.88 \ 10^{-3}s and CFL=1. First time step \Delta t=1.6 \ 10{-4}s is equal to CFL=2.84 \ 10^1, mesh size is equal to 512\!\times\!8. Second-order temporal convergence is confirmed for pressure, velocity, and density, for both L_2 and L_{\infty} norms.
Time step | Velocity L2 error | order | Velocity Linf error | order | Pressure L2 error | order |
---|---|---|---|---|---|---|
0.00016 | 0.04567511308746726 | n/a | 0.0969756971167619 | n/a | 18.419648481125442 | n/a |
8e-05 | 0.023040661811725035 | 0.987 | 0.05057052643421424 | 0.939 | 9.293373561480465 | 0.987 |
4e-05 | 0.007871063433196804 | 1.550 | 0.018461110879957804 | 1.454 | 3.1759544842508736 | 1.549 |
2e-05 | 0.0019860108340821712 | 1.987 | 0.00487317551443972 | 1.922 | 0.8016586080749134 | 1.986 |
1e-05 | 0.0003272307423698701 | 2.601 | 0.000751036560505236 | 2.698 | 0.13213976985276946 | 2.601 |
Time step | Pressure Linf error | order | Density L2 error | order | Density Linf error | order |
---|---|---|---|---|---|---|
0.00016 | 39.11237912552973 | n/a | 0.0001528094282489174 | n/a | 0.00032447634914145596 | n/a |
8e-05 | 20.392212028557736 | 0.940 | 7.70978394016906e-05 | 0.987 | 0.0001691738180567004 | 0.940 |
4e-05 | 7.451325473659168 | 1.452 | 2.634772261698379e-05 | 1.549 | 6.181620602019322e-05 | 1.452 |
2e-05 | 1.9683879274100207 | 1.920 | 6.6505608766816416e-06 | 1.986 | 1.632974885845684e-05 | 1.920 |
1e-05 | 0.3036241892607592 | 2.697 | 1.0962317060940545e-06 | 2.601 | 2.518866677236886e-06 | 2.697 |
We also present in the following tables the spatial convergence study with a constant Courant number CFL=1. Second-order spatial convergence is confirmed for all fields considering both L_2 and L_{\infty} norms.
Mesh | Velocity L2 error | order | Velocity Linf error | order | Pressure L2 error | order |
---|---|---|---|---|---|---|
16x8 | 0.05723446001301031 | n/a | 0.11153910125292044 | n/a | 23.083228485728153 | n/a |
32x8 | 0.030400313571282714 | 0.913 | 0.0684876275500017 | 0.704 | 12.263219838851635 | 0.913 |
64x8 | 0.010888678907157954 | 1.481 | 0.02650116841507552 | 1.370 | 4.394197978233194 | 1.488 |
128x8 | 0.0028515391729985223 | 1.933 | 0.006872393102497265 | 1.947 | 1.1511817510580113 | 1.933 |
256x8 | 0.0005231376899341912 | 2.446 | 0.0011813028362184785 | 2.540 | 0.2112741914471085 | 2.444 |
512x8 | 0.00013816071940312186 | 1.921 | 0.000283153575332179 | 2.061 | 0.05567420256992179 | 1.922 |
Mesh | Pressure Linf error | order | Density L2 error | order | Density Linf error | order |
---|---|---|---|---|---|---|
16x8 | 43.78283440304564 | n/a | 0.00019149849415735966 | n/a | 0.0003632224523231198 | n/a |
32x8 | 28.04361310572598 | 0.643 | 0.00010173568806080976 | 0.913 | 0.00023264985154902718 | 0.643 |
64x8 | 10.671073267337768 | 1.394 | 3.645427226010089e-05 | 1.481 | 8.852723799002149e-05 | 1.394 |
128x8 | 2.767924371417621 | 1.947 | 9.550205334804935e-06 | 1.932 | 2.296270425938829e-05 | 1.947 |
256x8 | 0.47675153113753765 | 2.537 | 1.7527309726821905e-06 | 2.446 | 3.955131335109385e-06 | 2.537 |
512x8 | 0.11399844733234943 | 2.064 | 4.6187325841851364e-07 | 1.924 | 9.457312704075349e-07 | 2.064 |
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