A Central-Upwind Scheme for Fluid Flows in a Nozzle With Discontinuous Cross-Section

A Central-Upwind Scheme for Fluid Flows in a Nozzle With

by

A Central-Upwind Scheme for Fluid Flows in a Nozzle With Discontinuous Cross-Section

ABSTRACT

The numerical solution of nozzle flow equations with discontinuous cross-sectional area is carried out in this project. The equations form set of complicated non-linear PDEs. The considered model can be used to describe several scientific and engineering problems. The exact solution of this set of equations is not available in the literature because of the stiff source term. Therefore we look for numerical methods for the solution of this type of equations. Here, the second order CUP scheme is implemented to simulate the given model Eqs. The legitimacy of the scheme is checked by taking some tough problems from literature. The results obtained by CUP scheme are compared with the results obtained from Central (NT) scheme.

Introduction

This post gives motivation about the study of fluid flow in a nozzle with discontinuous cross-section, describe literature review, discuss problems that occur when they are resolved numerically, provides a summary of the numerical scheme suggested and provides a summary of the content of this thesis.

Motivation

A Nozzle is a device often used to control the speed, rate of flow, pressure, direction and mass. It is considered to be a tube with cross sectional area to direct fluid flow. To increase the fluid kinetic energy a nozzle can also be used at the cost of pressure and inertial energy. One can classify nozzles into
divergent and convergent nozzles. To speed up the subsonic fluids convergent nozzles are used and to slow down the fluids divergent Nozzles are used if their flow is subsonic. It became an active field of current research.

There are a large number of present-day engineering devices in activity that depend to a limited extent, or altogether, on the comprehension of fluid dynamics planes in flight, ships adrift, autos out and about, mechanical biomedical devices, etc. The work on fluid dynamics were started by Aristotle. He given the concept of continuum and resistance. A major contribution of the Greek scientist. In the work of fluid dynamics, he gives the concept of pressure in fluid statics. After this Leonardo da Vinci interested to give the concept of continuity “AV=Constant” in 15th century for in-compressible fluid. Here ‘A’ shows cross-sectional area and ‘V’ is the velocity of the fluid.

A Central-Upwind Scheme for Fluid Flows in a Nozzle With Discontinuous Cross-Section

Toward the start of the nineteenth century, the equations of fluid motion as determined by Euler were notable. Despite their value in the 18th and 19th centuries, these equations significantly overlooked friction, a physical phenomenon often ignored in theoretical analysis. The Navier-Stokes equations, on the other hand, include terms to account for friction. The recent work in the field of shallow water wave equation and Euler equation with compressible and incompressible fluid flows.

Literature Review

In recent times, researchers have conducted numerous studies on the noise behavior of explicit hyperbolic equations. These studies include frameworks for balance laws in non-conservation laws and coupled Euler frameworks with distinct pressure laws. Coquel et al. developed entropy-satisfying numerical schemes for flows in nozzles with discontinuous cross sections. Naryan et al. published an analysis of the De-Laval nozzle using ANSYS Workbench. Coquel et al. worked on the entropy-satisfying numerical schemes for flows in nozzles with discontinuous cross section. Recently, the analysis of De-Laval nozzle on ansys workbench is published by Naryan et al.

Coquel et al. worked on the numerical calculation of the solutions of 1-D barotropic flows in a nozzle. These authors introduce the definition of shock waves and highlight the simple definition in the nonlinear field. We performed numerical calculations for Cauchy problems involving one-dimensional non-conservative hyperbolic systems and discussed the concept of weak solutions for these systems. D. Marchesin and P.J. Pase-Leme worked for several solutions in the Riemann problem for a varied elliptic-hyperbolic type system of two quadratic polynomial conservation laws and backing of the requirement for theoretical modification in the hypothesis of shock waves. The author studies about Riemann problem and generalized Riemann method.  Also works on Riemann problem but the system is inhomogeneous and not strictly hyperbolic. The exact Riemann problem for Euler equations in a vessel with discontinuous cross-section.

A Central-Upwind Scheme for Fluid Flows in a Nozzle With Discontinuous Cross-Section

Basic methodology and the main ingredients used in schemes are provided in this post and numerical results are more accurate than the standard well-balanced approach. The  works on Cauchy problem for the 2×2 non-strictly hyperbolic or hyperbolic and introduced new method Generalized Riemann Solution. Perthame and Simeoni used upwind basis at interface method presented is basically first order accurate. Applied numerical schemes for weak solutions of 10-moment Gaussian.

Numerical Challenges

The present work is related to the numerical study of one dimensional fluid flow in a nozzle with discontinuous cross-section area. The system of equations are non-conservative and contains stiff source term. The most challenging task is the discretization of source term and obtain the solution.

To develop robust, effective and accurate scheme which can solve the Nozzle flow model equations, several numerical complexities arises.
Few of them are listed as:

(1): How to handle the stiff source term and how accurately we discritize this term in nozzle flow model equations.

(2): How accurately approximation can be done.

(3): How to simulate the balances laws while maintaining the steady-state conditions.

Proposed Numerical Technique

This post is related to the numerical solution of nozzle flow equations. The model forms a complicated set of nonlinear PDEs. Our motivation is to extend an, accurate, effective numerical scheme for simulating the considered equations. We implemented and formulated the CUP scheme. To evaluate its efficiency, we tested it on challenging problems from the literature.

Central Upwind (CUP) Scheme

Central schemes are amongst those numerical schemes to solve several engineering and scientific problems which avoid eigen-structure of the problem. The backbone of such schemes is Lax-Friedrichs scheme of order one. The central scheme (NT) by Nessyahu-Tadmor is a Riemann-solver-free which does not require information about eignstructure. However, there are some drawbacks of these schemes. One of the them is for a small time step it produces numerical diffusions in the solution profiles. To handle this drawback, Kurganov and Tadmor developed the semi-discrete CUP scheme by utilizing local propagation speed. This scheme of kurganov and Tadmore is effective, robust and simple to implement. Furthermore, CUP scheme reduces the numerical dissipation usually observed central (NT) schemes.

The shallow water magnetohydrodynamic equations (SWMHD)

In this post we have reviewed the article of Qamar et al. This post includes a detailed derivation of the mathematical model and a 2D CUP scheme. We also conducted several test cases to evaluate the CUP scheme’s performance.

The SWMHD Equations

In this section, derivation of the SWMHD equations using ideal MHD equations is included. This derivation is analogous to the derivation by Gilman and Rossmanith.

We start with the ideal MHD equations alongwith gravitational force

(1)   \begin{align*} \partial_t\begin{bmatrix}\rho\\\rho \bf{v} \\\mathcal{E}\\\tilde{\bf{B}}\end{bmatrix}+\nabla\cdot\begin{bmatrix}\rho \bf{v}\\\rho \bf{v} \otimes \bf{v} + \tilde{p} I -\tilde{\bf{B}} \otimes \tilde{\bf{B}} \\\bf{u}(\mathcal{E} + \tilde{p}) - \tilde{\bf{B}} (\bf{v}\cdot \tilde{\bf{B}})\\\bf{v}\otimes \tilde{\bf{B}} - \tilde{\bf{B}} \otimes \bf{v}\end{bmatrix}&=\begin{bmatrix}0\\-\rho g \,\hat{\bf k}\\0\\0\end{bmatrix}\,\end{align*}

(2)   \begin{align*}\nabla \cdot \tilde{\bf{B}}& = 0,\end{align*}

Where, \bu=(v_1,v_2,v_3)^T, \bB=(B^{}_1,B^{}_2,B^{*}_3)^T, \gamma, \rho, p, \mathcal{E}, \tilde{p}, represent the velocity, magnetic field, ideal gas constant, density, thermal pressure, magnetic pressure, the total pressure, and the ideal gas constant, respectively. As the number of equations and number of unknowns are not equal so the system is not closed. In order to close the system we need some extra relations hich are called closures. Moreover, g is gravitational constant and \hat{\bf k} is outward normal vector. The fluid is balanced in the vertical direction, megnetohydrostatically.

From 1 the mass conservation law can be written as

(3)   \begin{align*} \partial_t \rho + \nabla \cdot (\rho \bf{v})&= 0, \end{align*}

where \rho = 0, \nabla = (\partial_x,\partial_y,\partial_z), and \partial_t \rho= 0. With these above assumptions equation 3 reduces to

(4)   \begin{align*} \partial_x v_1+ \partial_x v_2+\partial_x v_3&= 0\,, \end{align*}

On the same pattern, the momentum equations in 1 are

(5)   \begin{align*} \partial_t v_1+ \partial_t[{v_1}^2-{B^{}1}^2+gh+gb]+\partial_y[{v_1}{v_2}-{B^{}_1}{B^{}_2}]+\partial_z[{v_1}{v_3}-{B^{}_1}{B^{}_3}]&=0, \end{align*}

(6)   \begin{align*} \partial_t v_2+ \partial_x[{v_2}{v_1}-{B^{}_2}{B^{}_1}]+\partial_y[{v_2}^2-{B^{}_2}^2+gh+gb]+\partial_z[{v_2}{v_3}-{B^{}_2}{B^{}_3}]&=0, \end{align*}

(7)   \begin{align*} \partial_t v_3+ \partial_x[{v_3}{v_1}-{B^{}_3}{B^{}_1}]+\partial_y[{v_3}{v_2}-{B^{}_3}{B^{}_2}]+\partial_z[{v_3}^2-{B^{}_3}^2+gh+gb]&=0.\end{align*}


The magnetic field is expressed through the following equations

(8)   \begin{align*} \partial_t B^{}_1+ \partial_y[{v_2}{B^{}_1}-{B^{}_2}{v_1}]+\partial_z[{v_3}{B^{}_1}-{B^{}_3}{v_1}]&=0, \end{align*}


(9)   \begin{align*} \partial_t B^{}_2+ \partial_x[{v_1}{B^{}_2}-{B^{}_1}{v_2}]+\partial_z[{v_3}{B^{}_2}-{B^{}_3}{v_2}]&=0, \end{align*}


(10)   \begin{align*} \partial_t B^{}_3+ \partial_x[{v_1}{B^{}_3}-{B^{}_1}{v_3}]+\partial_y[{v_2}{B^{}_3}-{B^{}_2}{v_3}]&=0. \end{align*}

A Central-Upwind Scheme for Fluid Flows in a Nozzle With Discontinuous Cross-Section

Numerical Tests

This test case study is used to authenticate the efficiency of suggested CUP numerical scheme. For this purpose we take the following advection equation

(11)   \begin{align*}w_t+w_x=0,\end{align*}


at t=0, w= \sin(\pi x), 0 \le x\le 2. By using the method of characteristics, the exact solution is obtained and is expressed as

(12)   \begin{align*}w(x,t)=\sin(\pi(x-t)).\end{align*}


Table given below depicts L^{1} errors among both the schemes along with experimental order of convergence (EOC). The following table shows that order of both the schemes is two. But the CUP has comparatively less L^{1}-error.

Sr No.Central Upwind (CUP)KFVS
L^1–error \qquad EOCL^1–error \qquad EOC
50
100
200
400
800
1600
3200		.0182 \qquad –
.00449 \qquad 2.02
.00111 \qquad 2.02
.000277 \qquad 2.002
.000069223 \qquad 2.003
.00001729 \qquad 2.001
.00000432 \qquad 2.0		.0365 \qquad –
.00904 \qquad 2.02
.00249 \qquad 1.86
6.7265\times 10^{-4} \qquad 1.89
1.768\times 10^{-4} \qquad1.93
4.575\times 10^{-5} \qquad1.95
1.179\times 10^{-5} \qquad1.95
Schemes comparison.

Numerical Solution of Nozzle low model Equations

This post is concerned with Nozzle flow with discontinuous cross section area and \left(CUP\right) scheme. We discuss the Model of the Nozzle flow and discuss the physical assumptions necessary to accomplish this. Once discuss the non-hyperbolic structure of the continuity equations.

A Central-Upwind Scheme for Fluid Flows in a Nozzle With Discontinuous Cross-Section

Brief Description of the Model

The Mathematical Structure of SW Equations

Nozzle flow equations describe several mechanical engineering and planetary systems. For example, nozzles are used in diesel engines and various astrophysical scenarios. The 1D nozzle shallow model Eqs. are as follows:

(13)   \begin{equation*}\frac{\partial\left(\mathrm{a_0}\rho_0\right)}{\partial\mathrm{t}}+\frac{\partial\left(\mathrm{a_0}\rho_0\mathrm{u_0}\right)}{\partial\mathrm{x}}=0,\end{equation*}

(14)   \begin{equation*}\frac{\partial\left(\mathrm{a_0}\rho_0\mathrm{u_0}\right)}{\partial\mathrm{t}}+\frac{\partial\left(\mathrm{a_0}\left(\rho_0\mathrm{u_0^2}+\mathrm{p_0}\left(\rho_0\right)\right)\right)}{\partial\mathrm{x}}=\mathrm{p_0}\left(\rho_0\right)\frac{\partial\mathit{a_0}}{\partial\mathrm{a_0}},\end{equation*}

(15)   \begin{equation*}\frac{\partial\mathrm{a_0}}{\partial\mathrm{t}}=0,x\in\mathrm{R}, t>0.\end{equation*}

Here, \rho_0 and \mathrm{u_0} represent the density and the particle velocity of the fluid, respectively. The pressure \mathrm{p_0}=\mathrm{p_0}\left(\rho_0\right) is given by

(16)   \begin{equation*}\mathrm{p_0}\left(\rho_0\right)=\kappa\rho_0^\gamma,1<\gamma<\frac{5}{3} \end{equation*}

In above \mathrm{a_0}\left(\mathrm{x}\right)>0 represent the discontinuous cross section area of the nozzle. For smooth solution \left(\mathrm{x},\mathrm{t}\right)\mapsto\left(\rho_0,\mathrm{u_0},\mathrm{a_0}\right), the above system is equivalent to:

(17)   \begin{equation*}\frac{\partial\left(\mathrm{a_0}\rho_0\right)}{\partial\mathrm{t}}+\frac{\partial\left(\mathrm{a_0}\rho_0\mathrm{u_0}\right)}{\partial\mathrm{x}}=0,\end{equation*}

(18)   \begin{equation*}\frac{\partial\mathrm{u_0}}{\partial\mathrm{t}}+\frac{\partial\left(\frac{u_0^2}{2}+\mathrm{h_0}\left(\rho_0\right)\right)}{\partial\mathrm{x}}=0,\end{equation*}

(19)   \begin{equation*}\frac{\partial\mathsf{a_0}}{\partial\mathrm{t}}=0\end{equation*}

where the function \mathrm{h_0} is define by \mathrm{h_0}'\left(\rho_0\right)= \frac{\mathrm{p_0}'\left(\rho_0\right)}{\rho_0}, thus

(20)   \begin{equation*}\mathrm{h_0}\left(\rho_0\right):=\frac{\kappa\gamma}{\gamma-1}\rho_0^\gamma-1\end{equation*}

A Central-Upwind Scheme for Fluid Flows in a Nozzle With Discontinuous Cross-Section

The 1-D SW Equation

In 1-D the system takes the form

(21)   \begin{equation*}\frac{\partial\mathrm{U_0}}{\partial\mathrm{t}}+\frac{\partial\mathrm{G_0}}{\partial\mathrm{x}}=\frac{\partial\mathrm{H_0}}{\partial\mathrm{x}}\end{equation*}

where conserved variable is represented by \mathrm{U_0}, physical flux by \mathrm{G_0}\left(\mathrm{U_0}\right) and source term by \mathrm{H_0}\left(\mathrm{U_0}\right). They are given by

    \[\mathrm{U_0}= \left[\begin{array}{c} a_{0} \rho_{0} \\ a_{0} \rho_{0} u_{0}\\a_{0}\\\end{array}\right]\]

    \[\mathrm{G_0}\left(\mathrm{U_0}\right)= \left[\begin{array}{c} \mathrm{a_0} \rho_0 \mathrm{u_0}\\\mathrm{a_0}\left(\rho_0\mathrm{u_0}^2+\mathrm{p_0}\left(\rho_0\right)\right)\\\mathrm{0}\end{array}\right]\]

    \[\mathrm{H_0}\left(\mathrm{U_0}\right)= \left[ \begin{array}{c}\mathrm{0} \\\mathrm{p_0}\left(\rho_0\right)\frac{\partial\mathrm{a_0}}{\partial\mathrm{x}} \\\mathrm{0}\end{array}\right]\]

If we solve the above system, It will be convenient to write \mathrm{U_0}= \left(\rho_0, \mathrm{u_0}, \mathrm{a_0}\right). Then

(22)   \begin{equation*}\partial_t\mathrm{U_0}+\mathrm{A}\left(\mathrm{U_0}\right)\partial_x\mathrm{U_0}=0,\end{equation*}


Now we find eigenvalues of the Jacobian matrix,

    \[\mathrm{A}\left(\mathrm{U_0}\right)=\frac{\partial\mathrm{G_0}\left(\mathrm{U_0}\right)}{\partial\mathrm{U_0}}=\left[\begin{array}{ccc}\mathrm{u_0} & \rho_0 & \frac{\rho_0\mathrm{u_0}}{\mathrm{a_0}}\\\mathrm{h_0}'\left(\rho_0\right) & \mathrm{u_0} & \mathrm{0}\\\mathrm{0} & \mathrm{0} & \mathrm{0}\end{array}\right]\]


This matrix is not Jacobian, then

(23)   \begin{equation*}\lambda_1:=\mathrm{u_0}-\surd\mathrm{p_0}'\left(\rho_0\right),\lambda_2:=0,\lambda_3:=\mathrm{u_0}+\surd\mathrm{p_0}'\left(\rho_0\right),\end{equation*}

A Central-Upwind Scheme for Fluid Flows in a Nozzle With Discontinuous Cross-Section

Conclusions

This post numerically investigates nozzle shallow flow equations with discontinuous cross-sectional areas. The equations are forming non-linear complicated set of PDEs. The exact solution of nozzle flow equations are not available in the literature. We implemented a 1D central-upwind (CUP) scheme to numerically simulate the given equations. To validate our results, we compared them to second-order central (NT) scheme using several 1D test cases from existing literature. We calculated the L^{1}- error between the two schemes. The comparison of 1-D test case problems produced comparable results. The CUP scheme produce less errors as compared with central (NT) scheme. In summary, the scheme was able to capture sharp discontinuous profiles accurately.

Impact of Inclined Magnetic Field and Activation Energy (Part 3)

A Central-Upwind Scheme for Fluid Flows in a Nozzle With Discontinuous Cross-Section

Leave a Comment