Application of Weighted Essentially Non-Oscillatory Scheme to Solve the Dusty-Gas Flow Model

A weighted essentially non-oscillatory (WENO) finite volume scheme is extended to approximate the model of dusty-gas flow numerically. We use a WENO scheme of the fifth-order for the spatial reconstruction and a monotone numerical Lax-Friedrichs flux to upwind the fluxes. A 3rd-order total variation diminishing (TVD) Runge-Kutta algorithm is to be utilized to march the solution in time. In low density flows, the proposed numerical scheme effectively handles the contact discontinuities and more interestingly it sustains the positivity of flow variables. Furthermore, the proposed numerical scheme shows no spurious oscillations where shock waves and interfaces interact. Several one-dimensional Riemann problems are considered to analyse the accuracy of suggested numerical scheme. For validation, we compare the solutions generated by WENO-scheme with the solutions generated by well known central-upwind scheme and analytical solutions available for the model of dusty-gas flow. Citation: Aslam RD, Ali A, Rehman A, Qamar S (2018) Application of Weighted Essentially Non-Oscillatory Scheme to Solve the Dusty-Gas Flow Model. J Appl Computat Math 7: 412. doi: 10.4172/2168-9679.1000412


The Model of Dusty-Gas Flow
The governing equations for 1-D, time dependent flow of dusty-gas mixture are given as [10] ∂ t ρ+u∂ x ρ+ρ∂ x u=0 ρ(∂ t u+u∂ x u)+∂ x p=0 ∂ t +u∂ x p+ρc 2 x u=0 (1) where u represents the particle velocity along x-axis, the time is represented by t, the density is represented by ρ and the pressure is represented by p. The system in eqn. (1) agrees the Mie Gruneisen type equation of state which is expressed as [10] ( ) Where the quantities M and k p represents the volume fraction and the mass fraction respectively, R represents the gas constant as well as T represents the temperature of mixture. The relation among the quantities M and k p is given as Where ρ sp represents the specific density of solid-particles. The internal energy/unit mass for the mixture is defined by Furthermore the quantities c vm and c pm represents the specific heats of the dusty-gas mixture at constant volume and constant pressure respectively. Also, the quantities c v and c p represents the specific heat of gas at constant volume and pressure respectively. The quantity c sp denotes the speci c heat of solid-particles. The system of governing in eqn. (1) with eqn. (4) in conservative form are given as [17] ∂ t ρ+∂ x (ρu)=0 The system of eqns.

Exact Solution for the Model of Dusty-Gas Flow
A short summary of exact Riemann solver for the model of dustygas flow is presented here. For further details about the exact Riemann solver [17][18][19][20]29,30] and references there in. The exact solution to the Riemann problem in eqns. (6,10) ( ) has three waves associated with the eigenvalues in eqn. (9). These three waves divide the structure into four different constant states, and that are defined from left to right as: w L (known left data), w L * (unknown left data), w R * (unknown right data) and w R (known right data). The key step to solving the Riemann problem is finding the constant states w L * and w R * in the star region. In short, we need to evaluate the following physical quantities p L * =p R * =p * ,u L * =u R * =u * ,ρ L * and ρ R * . First we have to solve the non-linear algebraic equation for the pressure p * to find all these quantities. The algebraic equation for p * is given by Here u L is the velocity on left state and u R is the velocity on right state, ϕ L and R are functions of connecting the left and right states respectively, to the unknown regions. The function ϕ L is given by and the function ϕ R is given by The constant quantities A L , A R , B L and B R yields Here, Unlike the perfect gas dynamic, the solution within the rarefaction waves cannot be evaluated directly in case of dusty-gas flow. Therefore, we require an additional iteration procedure to evaluate the roots of nonlinear equation inside the rarefaction fan. This nonlinear equation with unknown p * for the left rarefaction is given by and the nonlinear equation for right rarefaction is shown as The unknown value of pressure p * in eqn. (11) or in eqn. (13) and (14) is obtained by Newton Raphson iterative procedure and the initial guess p o to start a iterative procedure is obtained by using the arithmetic mean of the initial data for pressure, as Once the corresponding to m different candidate stencils The coeffcient k nl gives guarantee that each m reconstructed values is m th order accurate, for detail [24]. For m=3in eqn. (20) becomes The non linear weights w 0 , w 1 and w 2 in eqn. (19) are defined as°( where γ m represents the linear weights, ∊=10 −6 and the smooth indicators ɸ m for the 5 th -order WENO finite-volume scheme are defined as The linear weights used in eqn. (23) are given as The semi-discrete scheme in eqn. (17) is re-written as In order to carry out third-order accuracy in temporal discretization, the 3 rd -order TVD Runge Kutta time discretization scheme [31], for more details [32][33][34], is applied to solve in eqn. (26) as follows where L(w) is the spatial operator and .

Numerical Test Problems
In this section, five 1-D numerical test problems are presented. The comparison of results obtained by WENO finite-volume scheme, central upwind schemes and the exact Riemann solver is given. As mentioned before, the solid-particles occupies only less than 5% of the total volume of mixture. Therefore, we need to choose the befitting value of value of pressure is known, then remaining unknown values are easily found by following the procedure as described for the ordinary gas dynamics [20].

Construction of WENO Finite-Volume Scheme for the Proposed Model
This section presents the construction of higher order WENO method finite-volume for the model of dusty-gas flow. The eqn. (6), we have The computational domain Ω is discretized with cells where ( ) ( ) . In order to obey upwinding for stability, we replace In eqn. (17) ( ) where w 0 , w 1 and w 2 are non-linear weights. In eqn. (18),
The purpose of this problem is to access the entropy satisfaction property of the suggested numerical scheme. The computational domain is used to be [0,2] at t=0.12 as final time and the numerical results are exposed in Figure 1. The comparison shows that both WENO scheme and central-upwind scheme give correct positions of discontinuities and an oscillation free pressure solution can be observed. However, it can be seen that WENO scheme resolves sharp discontinuities better than central-upwind scheme. Also the results computed by the WENO scheme are very close to the results of exact Riemann solver.

Problem 2:
A 123 test problem [36], that is also known as internal energy and low density problem. The problem having the following initial states ,0 1.0, 2,0.4 , 0.5 1.
This numerical test problem is used to analyze the ability of suggested numerical scheme for resolving contact discontinuity at low density. The solution profiles in computational domain [0,1] at t=0.15 are exposed in Figure 2 show that the suggested numerical scheme is more accurate than that of central-upwind scheme in term of resolving the contact discontinuity. The solution consist of two strong rarefaction wave thats why this problem is also known as strong rarefaction problem.
This test problem was introduced by the Woodward and Colella [37]. The pressure of the mixture is larger on the left data state than those of right data state. The main objective of the problem is to analyze the accuracy and robustness of the suggested numerical scheme. The computational domain used [0,1] is discretized into 200 mesh cells. The numerical results at t=0.012 displayed in Figure 3 show that there is a good agreement between the results generated by the WENO scheme and the exact Riemann solve. But we observe that central upwind scheme does not resolve contact discontinuity sharply in density profile as compare to WENO scheme.

Problem 4:
The collision of two shocks problem. The initial state is given below In this test problem we set ɸ=0.001667, the computational domain     Figure 4 show that the suggested scheme resolve the contact discontinuity superior than those of central upwind scheme. Also the results evaluated by the suggested scheme are very close to the results of exact Riemann solver. The solution incorporates the two strong shocks and a contact discontinuity.

Problem 5:
The Stagnant contact discontinuity problem. The initial state is given below In this test problem density on left data state is larger than those on right data state while pressure and velocity is same on both states causing a uniform constant state and there is no oscillation. In density profile, we observe that the stagnant contact discontinuity is sharply captured by the WENO scheme than those of central-upwind scheme displayed in Figure 5.

Conclusions
In this paper, the WENO finite-volume scheme was utilized for model of dusty-gas flow to get the numerical results. The fifth-order WENO scheme was used for the spatial reconstruction and for the temporal discretization. The 3 rd -order TVD Runge-Kutta algorithm was employed. Different test problems were considered to analyze the performance of proposed numerical scheme. The results obtained by suggested numerical scheme were compared with those of exact solutions and the central upwind scheme. The proposed numerical scheme preserved the non-oscillatory property near strong discontinuities. We observed that the WENO scheme efficiently resolved the contact discontinuities and most importantly it preserved the positivity of flow variables. Comparatively, the proposed numerical scheme produced better results than the central upwind scheme. An amazing agreement was noted among the solutions of suggested numerical scheme and exact Riemann solver.