Simulation of the mixture flow over terrain wave

. We work with the system of equations describing the non-stationary compressible turbulent multi-component ﬂow in the gravitational ﬁeld. We assume the mixture of perfect inert gases. The ﬂow over rough terrain is simulated with the use of the ﬁnite volume method. The modiﬁed Riemann problem is solved at the boundary faces. The roughness of the surface is simulated using the alteration of the speciﬁc dissipation at the wall, and with the use of the wall functions. Velocity proﬁle is compared with the experimental results. Own-developed computational code is used.


Introduction
Our aim is to numerically simulate the flow of the perfect gas mixture.We consider the viscous compressible gas flow, described by the Reynolds-Averaged Navier-Stokes equations with the k-ω model of turbulence.This system is equipped with the equation of state in more general form, and with the mass conservation of the additional gas specie.The expected result is to construct a method suitable for the estimates of the pollutant concentrations caused by the given source.This contribution follows previous work [1][2][3][4].

Formulation of the Equations
We consider the conservation laws for viscous compressible turbulent flow of ideal gas with the zero heat sources in a domain Ω ∈ I R N , and time interval (0, T ), with T > 0. The system of the Reynolds-Averaged Navier-Stokes equations in 3D has the form (1) Here x 1 , x 2 , x 3 are the space coordinates, t the time, r=1 τ sr v r + C k ∂θ/∂x s ) T are the viscous fluxes, S are additional sources.u = (v 1 , v 2 , v 3 ) T denotes the velocity vector, is the density, p the pressure, θ the absolute temperature, E = e+ 1  2 u 2 the total energy.Further τ i j = (µ+µ T )S i j −δ i j where µ * e-mail: kyncl@vzlu.cz is the dynamic viscosity coefficient dependent on temperature, µ T is the eddy-viscosity coefficient.For the specific internal energy e = c v θ we assume the caloric equation of state e = p/ (γ − 1), c v is the specific heat at constant volume, γ > 1 is called the Poisson adiabatic constant.The constant C k denotes the heat conduction coefficient C k = ( µ P r + µ T P r T )c v γ, and P r is laminar and P r T is turbulent Prandtl constant number.In our application of flow in the gravitational field we set the source terms to S = (0, g 1 , g 2 , g 3 , g • u), where g = (g 1 , g 2 , g 3 ) is the gravity vector.For the gas mixture with two species we use the Dalton's law for the total mixture pressure where p 1 and p 2 are the partial pressures of the first and second component gas.Let 1 and 2 be the mass density of these components.Then the total mass density of mixture is Temperature θ is same for all gases in the mixture, and the equation of state holds where R g = 8.3144621 is universal gas constant, and m i denotes the mollar mass of the ith specie.We can introduce the species mass fractions The thermodynamic constants of the mixture satisfy (using the decomposition of the internal specific energy and enthalpy) then the adiabatic constant γ, needed in the solution of (1), can be written as The system (1) is then extended with the conservation law of the mass for one gas component (specie) ( Here σ C is diffusion coefficient.The mass conservation for the second specie is automatically satisfied via the system ( 1).

Example: 3D diffusion of the passive mixture
Here we present a simple case for the test of the used numerical method.Let us assume the stationary flow with u = (0, 0, 0), and with constant σ = σ C µ T = 1.Let us solve the problem (2) with the boundary condition and initial condition given at time t 0 = 0.001 s The analytical solution is known and it can be written as The figure 1 shows computational results for unit domain, and comparison with analytical solution.

Example: the flow over terrain wave
We chose to simulate the flow over the rough terrain with the sinus wave geometry, accordingly to experiments shown in [12].Lenght of the wave was set to 0.800 m, height to 0.114 m.  as turbulence generator.The horizontal velocity profile U(y) in the start of the test section corresponds with its typical logarithmic shape estimated by the law of the wall (see [3,5]) where z S is the wall roughness, y denotes vertical coordinate, and u τ is the friction velocity, κ ≈ 0.41 (Karman's constant).The example of measured velocity profiles is shown in figure 2. The least squares method was used for the estimation of the values for u τ and z S .
For the simulations of the compressible gas flow in the atmosphere we assume the change of variables with the vertical coordinate y.We use the following initial conditons for the horizontal velocity component u, temperature θ, pressure p, density θ 0 = θ 00 − 0.0065y, Here θ 0 denotes total temperature, p 0 total pressure, θ 00 = 293.15K is the chosen total temperature and p 00 = 101325 Pa total pressure at y = 0 m, u ∞ = 10 m • s −1 is chosen velocity regime, and R = 287.04J • kg −1 • K −1 is the gas constant.Here we used the law of wall and the adiabatic approximations for perfect gas.
In order to to get the state variables at each boundary face we solve the local boundary problem with the use the original analysis of exact solution of the Riemann problem.This approach was shown and described in [13,14] and analyzed also in [8], [15][16][17].Using the thorough analysis of the Riemann problem we have shown, that the missing initial condition for the local problem can be partially replaced by the suitable complementary conditions.We suggest such complementary conditions accordingly to the desired preference.This way it is possible to construct the boundary conditions by the preference of total values, by preference of pressure, velocity, mass flow, temperature.On the contrary to the initial-value Riemann problem, the solution of such modified problems can be written in the closed form for some cases.Moreover, using such construction, the local conservation laws are not violated.
Here we used the boundary conditions by preference of known velocity at the outlet, as described in [13,15].This boundary condition was modified at the inlet by adding the known (estimated) total temperature, shown above.
For the simulation of the rough wall we use either the modification of the specific turbulent dissipation ω or the so-called wall functions for the estimation of values at the first cell (near wall).We estimate the friction velocity u τ , assuming the law of the wall (see [3,5]) to be valid at the closest volume, i.e. solve the equation where U P is the horizontal velocity at the considered cell, and y P denotes the cell center distance from surface.The wall friction τ w = µ ∂U ∂y | w is then estimated as τ w = w u 2 τ .
The momentum equations are solved with the modified effective viscosity µ e at the wall: µ e U P y P = τ w .The values for the k, ω at the first cell are set using log layer equations We use further modifications of this wall function shown in [4] in order to avoid the problem with possible zero velocity U P .
The regime constants for the considered flow above the rough terrain wave were chosen as

Conclusion
This paper is focused on the numerical simulation of the mixture of two inert perfect gases in the gravitational field.
The finite v olume m ethod i s a pplied f or t he s olution of the system of equations.The modification of the Riemann problem and its solution was used at the boundaries.All codes were implemented into the own-developed software.
The numerical examples of the flow above the rough surface simulating terrain wave were presented.

Fig. 1 .
Fig. 1. 3D diffusion: simulation and comparison with the analytical solution at time t = 0.01 s, x 1 -cut through the center.

uFig. 3 .
Fig. 3.The flow over the terrain wave: measured horizontal velocity profile at given vertical lines, and numerical simulation.Direct comparison is shown in the bottom picture.

Fig. 4 .
Fig. 4. The flow over the terrain wave: numerical simulation, mass fraction Y 1 of the emmision for various source placement.