numerical tests of far-field boundary conditions … · ows. the e ect of arti cial far- eld...

NUMERICAL TESTS OF FAR-FIELD BOUNDARY CONDITIONSFOR STABLY STRATIFIED FLOWS

T. Bodnar 1,2, Ph. Fraunie 3, H. Reznıcek 1,2

1 Faculty of Mechanical Engineering, Czech Technical University in Prague,Karlovo Namestı 13, 121 35 Prague 2, Czech Republic

2 Institute of Mathematics, Czech Academy of Sciences,Zitna 25, 115 67 Prague 1, Czech Republic

3 Mediterranean Institute of Oceanography - MIO, UM 110 USTV - AMU - CNRS/INSU 7294- IRD 235, Universite de Toulon, BP 20132 F-83957 La Garde cedex, France

Abstract

This numerical study presents the results of simulations of stably stratified wall-boundedflows. The effect of artificial far-field boundary conditions is studied in detail. The standardhomogeneous Neumann condition for pressure is replaced by a non-homogeneous conditiondepending on local velocity and its gradient. The two-dimensional tests are performed forthe case of flow over a low isolated hill. The simulations on computational domains withthree different heights are discussed to evaluate the performance of the new far-field artificialboundary condition. The model is based on Boussinesq approximation of non-homogeneousNavier-Stokes equations, solved using artificial compressibility method, looking for a steadysolution.

Keywords: Boussinesq approximation, stable stratification, far-field boundary condition, artificialcompressibility, finite-difference, finite-volume

1 Introduction

The numerical simulations of stably stratified flows involve many specific problems. As we haveshown e.g. in [4], [5], one of the major difficulties is related to boundary conditions used inmathematical modeling of this class of problems. Due to stable stratification, any local verticalflow perturbation (e.g. due to presence of an obstacle in the flow) leads to generation of massivegravity wave fields (see e.g. [1], [2]). These waves spread over large distances, much larger than thecharacteristic size of the original perturbation/obstacle. In case of homogeneous, i.e. non-stratifiedcase, the perturbations in the flow field usually vanish far from the obstacle, leading to unperturbedflow in the far-field. This situation dramatically changes in the presence of stratification, wheneven at very large distances from the obstacle, the flow field is still perturbed by the gravity waves.Technically it’s often impossible, or at least undesirable and expensive, to extend the size of thecomputational domain beyond the range of the gravity wave field. Thus it’s still necessary todeal with truncated computational domains, where the far-field can no longer be considered asundisturbed or fully developed. Thus the correct simulation of stably stratified flows is heavilydependent on the far-field boundary conditions. Most of the classical approaches used in non-stratified case will either completely fail or lead to unrealistic or inaccurate solution.

In this context, the work presented hereafter is a direct follow-up and extension of the of theboundary conditions review and discussion reported in [5], where also a new far-field pressureboundary condition was proposed.

2 Mathematical model

The full mathematical model describing the flow of variable density incompressible viscous (New-tonian) fluid flow is referred to as the non-homogeneous Navier-Stokes equations. It can e.g. befound in the classical book [8]. The model used for the present study is a bit simplified, assumingthe so called Boussinesq approximation.

TOPICAL PROBLEMS OF FLUID MECHANICS 17 _______________________________________________________________________

DOI: https://doi.org/10.14311/TPFM.2019.003

Governing equations

The Boussinesq approximation is obtained from the full system of non-homogeneous incompressibleNavier-Stokes equations by replacing the complete density ρ in the convective terms by a suitable(usually fixed in space and time) characteristic density ρ∗ and by subtracting the hydrostaticbackground part from the pressure. This immediately leads to the approximate set of governingequations, the so called Boussinesq approximation (see e.g. [4] for the derivation of the model):

divu = 0 (1)

∂ρ

∂t+ u · grad ρ = 0 (2)

∂u

∂t+ div(u⊗ u) =

1

ρ∗

(−grad p+ div2µD + (ρ− ρ0)g

). (3)

This is the system that was used to compute the fields of velocity u(x, t), pressure (perturbation)p(x, t) and density ρ(x, t) in the numerical simulations presented in this paper.

Far-field boundary conditions

The problem being discussed here is motivated by the stably stratified flow over an isolated hill.So it’s a wall bounded flow, where the main flow is parallel to the wall and is disturbed by a “wall-mounted” obstacle, which is in this case a smooth hill.The hill introduces a vertical perturbation(to the horizontal flow), which is responsible for generation of waves in the computational field. Thegravity waves have typically small amplitude, governed by the original (vertical) flow perturbation,while their wavelengths depend on the flow velocity and stratification magnitude (for given gravityacceleration). The typical wave field generated (in numerical simulation described further) by aflow over a hill is shown in the Fig. 1.

Figure 1: Vertical velocity contours and flow streamlines.

The flow structure shown in the Fig. 1 clearly demonstrates the typical wave motion of thefluid parcel along a streamline. It is manifested by a presence of quasi-periodic regions of ascend-ing/descending flow, i.e. positive/negative vertical velocity. Although the upper, far-field limit ofthe domain is placed at a distance corresponding to 20h, where h is the height of the obstacle(hill), the waves are still visibly present and the undisturbed flow is not recovered. When loweringthe upper boundary of the domain, the problem gets even more pronounced, so it doesn’t makesmuch sense to prescribe the far-field boundary conditions based on undisturbed flow. If for examplethe homogeneous Neumann condition is used for pressure1, i.e. ∂p

∂n = 0, the results shown in theFigs. 2, 3 and 4 will be obtained for horizontal velocity u, vertical velocity w and pressure p. Thecontour variables are nondimensionalized using the characteristic velocity U∗ and density ρ∗.

The Fig. 4 shows, that the homogeneous Neumann condition is respected at the upper boundary(i.e. the isolines are perpendicular to the boundary). But as a consequence, the vanishing vertical

1The boundary conditions discussed below are the numerical boundary conditions, needed on discrete level innumerical simulations (e.g. finite-difference or finite-volume) to fully determine the model and problem being solved.

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Figure 2: Contours of the non-dimensional horizontal velocity u = u/U∗.

Figure 3: Contours of the non-dimensional vertical velocity w = w/U∗.

Figure 4: Contours of the non-dimensional pressure p = p/(ρ∗U2∗ ).

pressure gradient force at the upper boundary leads to blocking of the vertical motion. Theresulting vertical velocity field shown in the Fig. 3 shows, that the contour patterns are clearlydisrupted close to the upper boundary. This is in contrast with the patterns shown for a thickerdomain in the Fig. 1. This homogeneous Neumann condition for pressure, that is often used in far-field for numerical simulations, heavily affects the solution and the results are, at least close to thefar-field artificial boundary, clearly non-physical. This can also be well seen from the comparisonof the enforced (by boundary condition) pressure field in Fig. 4 with the pressure Fig. 5 obtainedfor a larger domain (the same result shown in the Fig. 1).

Motivated by the above demonstrated failure of the homogeneous Neumann condition for pres-sure and by the previous tests summarized e.g. in [4, 5] a non-homogeneous Neumann condition

∂p

∂n= ρ∗|u|∂un

∂n, (4)

where un is the wall-normal component of velocity and |u| is the local velocity magnitude, isproposed and tested in the simulations below.


Figure 5: Pressure field obtained on a larger domain.

3 Numerical simulations

In order to evaluate the efficiency of the proposed boundary condition (4) a series of two-dimensionaltests was performed. The computational test case is motivated by the towing tank laboratory ex-periments described in [7] and numerical simulations from [6]. This is however just a starting pointand the actual configuration of our numerical test cases differs substantially in several aspects fromthe above mentioned papers.

The results presented in this paper were obtained using an in-house developed code. Thenumerical solver is based on a high resolution compact finite-difference space discretization andSSP Runge-Kutta time integration. For details see [1, 2, 3]. The sixth-order spatial discretizationwas combined with third-order SSP RK method in (pseudo-)time. The artificial compressibilitywas used to compute the pressure and to enforce the divergence-free constraint. In order to smooththe high frequency numerical oscillations, the eight order compact low-pass filter was used. As analternative, independent simulations were performed using a finite-volume code to confirm theresults. But hereafter, only the results from the finite difference solver are presented.

3.1 Computational setup

The two-dimensional computational domain was chosen as a rectangular part of a half-plane, withdimensions Lx×Lz. The z coordinate points in vertical direction (against the gravity acceleration)and the x coordinate is pointing in the free stream direction. The hill has a cosine shaped profilewith the height h and width 8h (see the schematic view in Fig. 6). The domain length, in terms of

8h

u=u(z)

u=UΩ

5h

8

h

Figure 6: Computational domain configuration.

maximum hill height h, is Lx = 60h. The hill is placed (together with the origin of the coordinatesystem) in the middle of the domain length, i.e. at the position Lx/2 = 30h from the inlet. Thethickness of the boundary layer, related to the inlet velocity profile, was set to 5h.

In order to be able to verify the influence of truncating the computational domain (vertically),three computational domains with the same surface profile, but different height were used. Thisis schematically shown in the Fig. 7. Besides of the standard height H = 10h, two extensions by50% each were used leading to heights 1.5H = 15h and 2.0H = 20h. The aim is to verify, how

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L

Lz

x

Ω

2H

H 1.5

H

Figure 7: Extension of the computational domain.

the truncation of the computational domain, can be compensated by using appropriate artificialfar-field boundary condition. So the solution is computed, using identical computational setup,on domains with three different heights. The mutual comparison between the results obtained onlarger and smaller domain shows to what extent the solution is degraded by the truncation of thecomputational domain.

Boundary conditions

The identical computational setup is used in all simulations2:

• Inlet . . . The velocity profile u = (u(z), 0, 0) is prescribed. The horizontal velocity componentu is given by the polynomial profile u(z) = U∗

(2z − z2

), where non-dimensional height z is

defined using the boundary layer thickness as z = z/(5h). Above the height z = 5h, theprofile is extended by the constant velocity U∞ = U∗. Density perturbation ρ′ = ρ − ρ0 isset to zero, i.e. ρ = ρ0(z). Homogeneous Neumann condition is used for pressure.

• Outlet . . . All velocity components and also the density (perturbation) are extrapolated.Pressure is set to a constant (zero).

• Wall . . . No-slip conditions are used on the wall, i.e. the velocity vector is set to u = (0, 0, 0).The density is extrapolated. Homogeneous Neumann condition is used for pressure.

• Far field . . . All velocity components and also the density are extrapolated. The non-homogeneouscondition (4) is used for pressure

In the simulations shown below, the fluid is characterized by the density ρ∗ = 1000 kg ·m−3and dynamical viscosity µ = 10−3kg ·m−1 · s−1. The linear background density profile is definedby ρ0(z) = ρ∗ + γ · z, with the (stable) vertical density gradient γ = −25 kg ·m−4. The gravityacceleration acts against the z coordinate, so g = −10m · s−2. The hill height was set to h =1cm = 0.01m and the velocity U∗ = 1 cm · s−1 = 0.01 m · s−1. These physical parameterscorrespond to the buoyancy (Brunt–Vaisala) frequency N =

√γg/ρ∗ = 1/2 , Reynolds number

Re = ρ∗hU∞/µ = 2 · 103 and Froude number F = U∞/Nh = 1.

3.2 Numerical results

The results are presented in the form of contours of velocity components. The Fig. 8 shows theresults on a full computational domain, for three different domain heights. In all cases, the wholecomputational domain is shown, up to the boundary, without any truncation or cut-off. Thisis essential to assess the effects of boundary conditions in the proximity of artificial boundariesof the computational domain. The same set of results is shown in the Fig. 9, but only thewindow corresponding to the smallest computational domain (with height H) is shown for easiercomparison. For reference, the solution obtained using the homogeneous Neumann condition forpressure on the smallest domain is shown in the Fig. 10.

The results obtained using the proposed non-homogeneous condition (4) are much better thanthose with the standard homogeneous Neumann condition. The isolines on truncated domain have

2Except for the homogeneous Neumann condition used for far-field pressure to obtain a reference solution.


horizontal velocity u = u/U∗ vertical velocity w = w/U∗

Figure 8: Velocity contours for domain height 2H (top), 1.5H (center) 1H (bottom).


Figure 9: Velocity contours for domain height 2H (top), 1.5H (center) 1H (bottom).


Figure 10: Velocity contours for domain height 1H - with the homogeneous Neumann condition.

similar shape as on the larger domain. The vertical motion is no more blocked by the artificialupper boundary.

To confirm this result, the whole above presented series of simulations was repeated with inletvelocity reduced to 1/2 of its original value, i.e. to U∗ = 0.005m/s. This leads to gravity waveswith wavelength also reduced to 1/2 of its original value. The corresponding results are presentedin the Figs. 11, 12, and 13.

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Figure 11: Velocity contours for domain height 2H (top), 1.5H (center) 1H (bottom)– reduced inlet velocity U∗ = 0.005m/s.


Figure 12: Velocity contours for domain height 2H (top), 1.5H (center) 1H (bottom)– reduced inlet velocity U∗ = 0.005m/s.


Figure 13: Velocity contours for domain height 1H - with the homogeneous Neumann condition– reduced inlet velocity U∗ = 0.005m/s.

4 Conclusions & Remarks

The above presented numerical simulations using the new non-homogeneous Neumann pressurecondition have shown very good performance, with minimum impact on the solution due to trun-


cation of the computational domain.

• Other boundaries – It is good to note that the new non-homogeneous pressure condition reducesto homogeneous one when either |u| = 0 (e.g. no-slip wall) or ∂un

∂n = 0 (e.g. on the inlet).This in fact means that the same condition for pressure can be (was) used on all boundaries,except the outlet where the pressure was set to a constant value.

• Finite-volume simulations – The finite volume simulations performed using the same model, onthe same grid provided almost identical results. The use of the condition on unstructuredgrids has to be tested. But because the condition has a form of non-homogeneous Neumanncondition, the implementation should be straightforward.

• Three-dimensional case – Three-dimensional simulations were performed with the same artificialpressure condition used on the sides of the domain. It performed equally well, withoutsignificantly affecting and degrading the solution.

The future investigation will focus on theoretical justification of the proposed condition and it’suse in the finite-element solution of stably stratified flows.

Acknowledgment

The financial support for the present work was provided by the Czech Science Foundation under the grant

No. P201-16-03230S. Tomas Bodnar is grateful for the invited professorship position he received at the

University of Toulon, which has made possible this joint work. The authors acknowledge the co-funding

from the Erasmus+ Programme of the European Union.

References

[1] T. Bodnar & L. Benes.: On some high resolution schemes for stably stratified fluid flows. InFinite Volumes for Complex Applications VI, Problems & Perspectives, volume 4 of SpringerProceedings in Mathematics, pages 145–153. Springer Verlag, 2011.

[2] T. Bodnar, L. Benes, Ph. Fraunie, & K. Kozel.: Application of compact finite-differenceschemes to simulations of stably stratified fluid flows. Applied Mathematics and Computation,219(7):3336–3353, 2012.

[3] T. Bodnar & Ph. Fraunie.: Numerical simulation of the wake structure behind three-dimensional hill under stable stratification. In Topical Problems of Fluid Mechanics 2016,pages 17–22, Prague, 2016. Institute of Thermomechanics CAS.

[4] T. Bodnar & Ph. Fraunie.: On the boundary conditions in the numerical simulation of stablystratified fluids flows. In Topical Problems of Fluid Mechanics 2017, pages 45–52, Prague,2017. Institute of Thermomechanics CAS.

[5] T. Bodnar & Ph. Fraunie.: Artificial far-field pressure boundary conditions for wall-boundedstratified flows. In Topical Problems of Fluid Mechanics 2018, pages 7–14, Prague, 2018.Institute of Thermomechanics CAS.

[6] L. Ding, R.J. Calhoun, & R.L. Street.: Numerical simulation of strongly stratified flow over athree-dimensional hill. Boundary-Layer Meteorology, 107(1):81–114, April 2003.

[7] J.C.R. Hunt & W.H. Snyder.: Experiments on stably and neutrally stratified flow over a modelthree-dimensional hill. Journal of Fluid Mechanics, 96(4):671–704, 1980.

[8] P.L. Lions.: Mathematical Topics in Fluid Mechanics, Volume I – Incompressible Models,volume 3 of Oxford Lecture Series in Mathematics and Its Applications. Oxford UniversityPress, 1996.

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