International Journal of Aerospace Engineering

Volume 2016, Article ID 5740435, 5 pages

http://dx.doi.org/10.1155/2016/5740435

## Numerical Investigation of Shock Wave Diffraction over a Sphere Placed in a Shock Tube

^{1}Moscow Aviation Institute, National Research University, Volokolamskoe Shosse 4, Moscow 125993, Russia^{2}Department of Mechanical Engineering, Ben-Gurion University of the Negev, 841050 Beer Sheva, Israel^{3}Peter the Great St. Petersburg Polytechnic University, Saint Petersburg 195251, Russia

Received 16 December 2015; Revised 21 April 2016; Accepted 19 June 2016

Academic Editor: Mohamed Gad-el-Hak

Copyright © 2016 Sergey Martyushov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

For evaluating the motion of a solid body in a gaseous medium, one has to know the drag constant of the body. It is therefore not surprising that this subject was extensively investigated in the past. While accurate knowledge is available for the drag coefficient of a sphere in a steady flow condition, the case where the flow is time dependent is still under investigation. In the present work the drag coefficient of a sphere placed in a shock tube is evaluated numerically. For checking the validity of the used flow model and its numerical solution, the present numerical results are compared with available experimental findings. The good agreement between present simulations and experimental findings allows usage of the present scheme in nonstationary flows.

#### 1. Theoretical Background

When a solid particle is exposed to a postshock gas flow, its response depends on the relative velocity that exists between the particle and the flow. Until the particle reaches a steady postshock flow velocity, the relative velocity between the particle and the gas flow changes and the particle motion is nonstationary. In shock tube experiments, the particle trajectory could be recorded accurately and its drag coefficient is evaluated from the recorded trajectory as follows. The equation of motion of a solid particle accelerated by the gas flow readswhere , , and are the solid sphere velocity, diameter, and material density, respectively. and are the gas velocity and density, respectively. It was shown in Igra and Takayama [1] that based on (1) the particle drag coefficient () and the appropriate Reynolds number can be expressed as follows:where and are components of the velocity vector in and directions, respectively, is the gravity acceleration, and is the gas viscosity. Similarly, the sphere’s Mach number, based on the relative velocity, readswhere and are the gas specific heat ratio and the gas constant, respectively. In the following, first the experimental results of Tanno et al. [2] are simulated. In their experiments a 80 mm sphere was kept on a strut in a 150 × 500 mm cross section shock tube. The case of free moving sphere will be considered in a separate paper; for such cases (1)–(3) are relevant. When the sphere is kept stationary throughout its collision with the oncoming shock wave, the drag coefficient is directly deduced from the computed pressure distribution around the sphere. In simulating the shock tube experiments the planar incident shock wave is initially situated at some distance upstream of the sphere. The main parameters controlling the flow are the sphere’s diameter, the cross section of the shock tube where the sphere is placed, the mass of the sphere, and the strength of the incident shock wave. As mentioned, in the following computations the sphere drag coefficient is deduced from the computed, nondimensional pressure coefficient around the sphere.

#### 2. Numerical Scheme

System of mass, momentum, and energy conservation equations for ideal nonviscous gas for a moving finite volume can be written in the following form:where , , is the vector of conserved variables, is vector of fluxes normal to the boundary of the control volume, , , , and is the velocity vector of the control volume boundary.

The explicit time step operator that represents a finite difference algorithm for approximating the system of (4) can be factored into a symmetric product of time step operators in directions. Vector of fluxes in a normal direction to the boundary of grid cell is determined on a basis of the finite difference scheme suggested by Yee et al. [3]. A slightly improved version of Harten’s scheme was suggested by Martyushov [4] and was used in the present calculations. These improvements are briefly outlined in the following. Vinokur [5] showed that Roe’s procedure employs the following formula for the sound velocity.

This expression may yield results outside the interval . In this case the procedure for calculating eigenvectors is incorrect and the sign of the eigenvalues can be wrong. In the present version of Harten scheme, this situation was taken into account and in the case when is outside the interval Roe interpolation for the values at boundary between cells is replaced by a half sum of the corresponding variables in the considered cells. For calculation of characteristic variables at the cell boundary using Harten’s scheme geometric and gas dynamics variables is used in five cells: . In order to reduce the influence of the geometric parameters of distant cells instead of the characteristic variables , , and in the present computation, pseudocharacteristic variables , , and are used. Detailed investigation of the algorithm and its validation for one-dimensional flows can be found in Il’in and Timofeev [6].

The numerical grid used in the present calculations was generated using the Thompson-type algorithm based on solving a system of three Poisson equations (see Thompson et al. [7] and Martyushov [8]). Example of the employed grid is shown in Figure 1 where every fifth coordinate line is plotted.