The high hall ventilation with the simplified simulation of the fan

Here we work with the system of equations describing the non-stationary compressible turbulent multi-component flow in the gravitational field. We focus on the numerical simulation of the fan situated inside the high hall. The RANS equations are discretized with the use of the finite volume method. The original modification of the Riemann problem and its solution is used at the boundaries. The combination of specific boundary conditions is used for the simulation of the fan. The presented computational results are computed with own-developed code (C, FORTRAN, multiprocessor, unstructured meshes in general).


Introduction
In this contribution we consider the Reynolds-Averaged Navier-Stokes equations (RANS) with the k-omega model of turbulence, shown in [1,2]. We focus on the real gas flow in the gravitational field. The special method for the simulation of the simple fan is presented, using two compatible boundary conditions, used previously in [3]. We focus on the numerical solution of these equations, using owndeveloped software.

Methods
For the discretization of the system we use either explicit or implicit finite volume method in order to discretize the analytical problem, represented by the system of equations in generalized (integral) form. In order to apply this method we split the area of the interest into the elements, and we construct a piecewise constant solution in time, as described in [4]. The crucial problem of this method lies in the evaluation of the so-called fluxes (and its Jacobians) through the edges/faces of the particular elements. We use the exact Riemann solver for the solution of the local problem at each face, with the theory shown in [5]. At the boundary faces it is necessary to solve the incomplete Riemann problem, where the right-hand side initial condition is not known. It can be shown, that this right-hand side initial condition for the local problem can be partially replaced by the suitable complementary condition. Various original modifications of the Riemann problem (and exact solutions of these modifications) were shown and analyzed in [6][7][8][9][10]. These boundary conditions were also used in [11,14,13]. Here we present the simulation of the fan, using two connected boundaries. At these boundaries we solve the conservation laws, using the modification of the Riemann problem by the preference of the total quantities at the inlet, and the modification by the preference of the mass flow at the outlet. Using such boundary conditions it a e-mail: kyncl@vzlu.cz is necessary to compute the solution of the resulting nonlinear problems. The combination of these two boundary conditions was used previously for the simulation of the propeller disk in [3]. The partial boundary condition with the preference of mass flow is sometimes being implemented with the use of some iterative process, guessing the correct values (for the pressure, density, velocity) in order to match the given mass flow through the boundary. In our approach we try to be as exact as possible, using our own original procedures. We follow the exact solution of the initial-value problem for the system of hyperbolic partial differential equations, the original analysis was shown in [14]. The own-developed software (C, FORTRAN) is based on the finite volume method with the implicit or explicit time discretization, solution is computed on unstructured 3D meshes in general. Multiple types of parallelization are used OpenMPI (for the computers with shared memory), MPICH (for the computers with the distributed memory), CUDA (graphic cards computations) are used, see also [15][16][17][18]. The large linear systems within the implicit method are solved with the implemented preconditioned generalized minimal residual (GMRES) method.

The Riemann problem for the split Euler equations
For many numerical methods dealing with the two or three dimensional equations, describing the compressible flow, it is useful to solve the Riemann problem for the 3D split Euler equations. We search the solution of the system of partial differential equations in time t and space (x, y, z) equipped with the initial conditions Vector u = (u, v, w) denotes the velocity, density, p pressure, E = e + |u| 2 is the total energy, with e denoting the specific internal energy. We assume the equation of state for the calorically ideal gas .
'Split' means here that we still have 5 equations in 3D, but the dependence on y, z (space coordinates) is neglected, and we deal with the system for one space variable x.
The solution of this problem is fundamentally the same as the solution of the Riemann problem for the 1D Euler equations, see [4, page 138]. In fact, the solution for the pressure, the first component of the velocity, and the density is exactly the same as in one-dimensional case. It is a characteristic feature of the hyperbolic equations, that there is a possible raise of discontinuities in solutions, even in the case when the initial conditions are smooth, see [19, page 390]. Here the concept of the classical solution is too restrictive, and therefore we seek a weak solution of this problem. To distinguish physically admissible solutions from nonphysical ones, entropy condition must be introduced, see [19, page 396]. By the solution of the problem (1)-(5) ,(6),(7) we mean the weak entropy solution of this problem in Q ∞ . The analysis to the solution of this problem can be found in many books, i.e. [4], [19], [5]. The general theorem on the solvability of the Riemann problem can be found in [4, page 88].
The solution is piecewise smooth and its general form can be seen in Fig. 1, where the system of half lines is drawn.
Using the theory in [4], [19], [5], we can write the solution for the primitive variables as The folowing relations for these variables hold: Here a L = γp L / L , a R = γp R / R , and γ denotes the adiabatic constant. Further s 1 T L denotes "unknown left wave speed", s 3 T R "unknown right wave speed". Note, that the system (8) -(13) is the system of 6 equations for 6 unknowns p ,u , L R ,s 1 T L ,s 3 T R .
Solution for the Pressure p The combination of the equations (8), (11) gives the implicit equation for the unknown pressure p This is a nonlinear algebraic equation, and one cannot express the analytical solution of this problem in a closed form. The solution p can be found as the root of the function F(p) defined as The analysis of this function can be found in [4]. Here we state, that F(p) is monotone and concave. Also the analytic expression for its derivative is available. For a positive solution for the pressure F(0) < 0 is required. This gives the pressure positivity condition The Newton method can be used to find the root of F(p) = 0. With the pressure p known, we use the equations (8)- (13) to obtain the whole solution.

Remarks
-Once the pressure p is known, the solution on the lefthand side of the contact discontinuity depends only on the left-hand side initial condition (6). And similarily, with p known, only the right-hand side initial condition (7) is used to compute the solution on the righthand side of the contact discontinuity. -The solution in Ω L ∪Ω HT L ∪Ω L (across 1 wave) (solvability in general case) The solution components in Ω L ∪ Ω HT L ∪ Ω L region must satisfy the system of equations (8)- (10). It is a system of three equations for four unknowns ( L , p , u , s 1 T L ). We have to add another equation in order to get the uniquely solvable system in Ω L ∪ Ω HT L ∪ Ω L . This is the key problem for the outlet boundary condition.
-The equation (8) can be written as the equation for pressure, see [7] p = P(u ), , In our work we use the analysis of this problem also for the solution of the initial-boundary value at the boundary faces.

Boundary condition by preference of total quantities and the tangential component of velocity
Using this boundary condition, we try to prescribe given total quantities at the face, comlete analysis was shown in [7], [20]. The conservation laws must be satisfied in close vicnity to the boundary face. We use the analysis of the incomplete Riemann problem to construct the values for the density, pressure and velocity. The following notation is used (some values are used only for an INLET case) At the vicinity of the boundary we solve the modified Riemann problem for the split Euler equations (1)-(5), introduced in Section 3, with the left-hand side initial condition (6) and the complementary conditions (prescribed direction of velocity and given total quantities).
Here θ o > 0, p o > 0 are given constants, R denotes the characteristic gas constant, and γ is the adiabatic constant. The equations (18), (19) are considered for the velocity u ∈ According to analysis shown in [20], there are two possibillities for the 1-wave, either there is a 1-shock wave and the sought velocity u can be found as a root of the function considered for u < u L , − 2a 2 o (γ−1) < u < 0, and with P S (u) defined in (16).
Or there is a 1-rarefaction wave, and the velocity u can be computed combining (18), (19), (16) as Summarizing the above possibilities of the 1-shock and the 1-rarefaction wave we can construct the algorithm in figure 2. for the computation of the velocity u in the star region. - ? no u = root of (21) Knowing the velocity u we compute the pressure p using (19), and the density R = p Rθ R . At last we set the boundary values as The problem (1)-(5) ,(6), (19), (18)  γ−1 a L }) ≤ 0 then the problem does not have a negative solution, and for the practical applications we choose the velocity u = min{0, u L + 2 γ−1 a L }, or we use the boundary condition preferring the static pressure p = p o , see [10,7], in this case.
Letp 0 ,θ 0 be the given total quantities together with the given tangential velocity components : v t and w t are the velocity components in the direction of o and p. Here o·n = 0, |o| = 1, and p = n × o. Then we set and we use the same algorithm (figure 2.) for the computation of the velocity u . Then it is p B = p , B = R , and u B = (n 1 u +o 1 v t +p 1 w t , n 2 u +o 2 v t +p 2 w t , n 3 u +o 3 v t +p 3 w t , ).

Outlet boundary condition by preference of massflow
Using this boundary condition, we try to prescribe given total quantities at the face, comlete analysis was shown in [8,14]. The conservation laws must be satisfied in close vicnity to the boundary face. We use the analysis of the incomplete Riemann problem to construct the values for the density, pressure and velocity. The following notation is used (some values are used only for an INLET case) At the vicinity of the boundary we solve the modified Riemann problem for the split Euler equations (1)-(5), introduced in Section 3, with the left-hand side initial condition (6) and the complementary conditions prescribing the mass flow where G ≥ 0 is given constant (G = mass f low f ace area ). We are interested in the solution with u > 0, and s L? < 0, which guarantees the possibility of the values to be prescribed at the boundary. In general, there are two possibilities of the wave pattern, which may interest us.

-1-shock wave
Here, we are interested in the solution with s 1 < 0, u > 0. Let us construct the function F S (u), using the relations (8),(9) where and P S (u) defined in (16). The sought velocity u is the root of this function F S (u). It is s 1 < 0 for u < u X = u L , with M 2 L = u 2 L /γp, see [7]. The first derivative is F S > 0 for u < min{u L , u X }. The problem has a solution with the 1-shock wave if F S (min{u L , u X }) > 0. Let us assume the solution with the 1-rarefaction wave. Using (8), (9),(23), the velocity u is the root of the function Derivative F R (u) > 0 in the interval (0, u 0 ) and F (u) < 0 for u ∈ (u 0 , u 1 ) . The maximum of the function F R is at the point u 0 . We are interested in the solution with s T L < 0. This is satisfied only fo u < u 0 .

Algorithm
Here we present the possible algorithm for the solution of the sought values at the boundary. The resulting boundary velocity in the global coordinates is u B = n(u B − u L ) + u L .

Simple fan simulation
The simple fan simulation is based on the boundary conditions shown above in sections 4, 5. The inlet boundary condition with the given total temperature and the total pressure is connected with the outlet boundary condition with the preference of the mass flow. This procedure was succesfully used by authors in [3].
Here we suppose, that the total quantities θ 0FAN , p 0FAN and the tangential velocity components at the fan outlet are known. Then we may use the boundary condition in section 4 to compute the boundary state (at the fan outlet). The conservation law of mass is valid across the fan, and therefore we may use the known computed massflow (at the fan outlet) to complement the incomplete the Riemann problem at the fan inlet. The solution to this problem gives us the boundary state at the fan inlet, analysis and the algorithm is shown in section 5.
The approach is based on the estimation of the total quantities θ 0FAN , p 0FAN and the tangential velocity at the fan, wich may be rather complicated. Let us for example suppose, that the distribution of the velocity u FAN is known at the fan outlet (at some time instant), together with the total quantities p 0X , θ 0X before the fan rotor. Then we may estimate .

Examples
Here we show the computational results for the flow around and inside the high hall in the isothermal atmosphere. The initial condition was constant in the whole domain with the total temperature θ 0 = 293. 15[K], zero velocity, and the total pressure distribution modified in the gravitational field as p 0 = p 00 e 9.

Conclusion
This paper shows the numerical simulation of the fan situated inside the high hall. The partial differential equations describing the conservation laws are solved numerically with the use of the finite volume method. Own software was programmed. The special modification of the Riemann problem and its solution was used at the boundaries. The combination of these boundary conditions was used for the simulation of the fan, which is the original result of this work.