# Numerical Simulation of Viscous Perfect Gas Dynamics

Vyacheslav Antonovich Bashkin
Central Aerohydrodynamic Institute (TsAGI), Zhukovsky, Moscow region, Russia

Ivan V. Egorov
Central Aerohydrodynamic Institute (TsAGI), 1 Zhukovsky str., Zhukovsky, 140180, Moscow Region, Russia; Moscow Institute of Physics and Technology (MIPT), 9 Institutsky per., Dolgoprudny, Moscow reg., 141700, Russia

## Descrição

The monograph summarizes the results of research of many years in the field of numerical simulation based on the continuum dynamics equations for transonic and supersonic flows of viscous perfect gas in the context of the problems of external aerodynamics, which have been obtained by the authors and their colleagues and published in different domestic journals. The monograph consists of two parts. The first part deals with stationary and non-stationary two-dimensional problems, stationary three-dimensional problems are considered in the second part. That said, the uniform steady flow over the bodies of comparably simple configuration, which surface is determined analytically, is studied. Each part begins with mathematical statement of the problem and method of its numerical analysis. This is followed by the detailed discussion of the results of computations for a number of aerodynamic problems, which have been obtained within some range of the key similarity parameters. The problems under consideration are divided into two groups according to the purposes. The problems intended to the theoretical study of the flow field near the streamlined body, its local and total aerodynamic characteristics and of the effect of the key similarity parameters on the flow field characteristics refer to the first group. For this purpose it is necessary to obtain computational results within a wide range of the key similarity parameters. Such research is usually performed using the bodies of simple configuration, with the great attention being paid to the verification of the numerical simulation method and adequacy of the results. The monograph considers the classic bodies such as: circular and elliptical cylinders, sharp circular and elliptical cones. The problems related to the computational validation of the aerodynamic experiment in different supersonic and hypersonic wind tunnels of TsAGI refer to the second group. In this case the computations are implemented for the considered body as applied to the experimental conditions. That said, in most cases the computations include estimation of the flow field along the whole channel of the wind tunnel, i.e. the flow of the actuating medium is calculated in the nozzle and in the test chamber of the wind tunnel both with and without the model in it. The results of computations are compared with the experimental data. The monograph considers mainly the axisymmetric blunted bodies of Martian probe type (United States and European analogs). The monograph is of interest for professionals in the field of computational and applied aerodynamics as well as for undergraduate and graduate students whose major is somehow related to the applied aerodynamics.

378 pages,
© 2016

## Table of contents:

Section 1. Numerical Simulation of Two-Dimensional Problems of External Aerodynamics

Chapter 1: Mathematical Problem Statement and Numerical Analysis

1.1.1 Differential Navier-Stokes Equations

1.1.2 Boundary and Initial Conditions

1.1.3 Differential Reynolds Equations

1.1.4 Boundary and Initial Conditions

1.2 Approximation of Equations

1.3 Solution of Nonlinear Finite-Difference Equations

1.4 Solution of Systems of Linear Algebraic Equations

1.4.1 Direct Method of Solution of a System of Linear Algebraic Equations

1.4.2 Iterative Method of Solution of a System of Linear Algebraic Equations

1.4.3 Acceleration of Convergence by Means of Preconditioning

1.5 Parametric Computations

1.5.1 On the Effectiveness of Numerical Solution of Finite-Difference Equations

1.5.2 On Solution Convergence in the Number of Computational Grid Nodes

Chapter 2: Circular Cylinder in Transonic Viscous Perfect Gas Flow

2.1 Circular Cylinder in Uniform Uncompressible Fluid Flow

2.2 A Circular Cylinder in Transonic Flow

2.3 Circular Cylinder at M<sub>∞</sub> = 0.8 and Re = 105

2.3.2 Evolution of Streamline Pattern

2.3.3 Evolution of Vorticity Field

2.3.4 Evolution of Pressure Coefficient Distribution along the Cylinder Surface

2.3.5 Evolution of Distribution of the Local Friction Drag Coefficient along the Cylinder Surface

2.3.6 Verification of Numerical Simulation

2.4 Circular Cylinder at Transonic Mach Number and Re = 105

2.4.2 Evolution of Gasdynamic Variables at the Reference Points of the Flow Field

2.4.3 Evolution of Pressure Coefficient at the Reference Points of the Cylinder

2.4.4 Verification of Numerical Simulation Data and Discussion

2.5 Circular Cylinder at Mach Number M<sub>∞</sub> = 0.8 and Different Reynolds Numbers

2.5.2 Aerodynamic Characteristics of the Cylinder

Chapter 3: Circular Cylinder in Supersonic Viscous Perfect Gas Flow

3.1 Verification of the Numerical Approach

3.1.1 Experimental and Numerical Conditions

3.1.3 Local Characteristics

3.1.4 Integral Characteristics

3.2 Instantaneous Launch with Supersonic Velocity

3.2.1 Evolution of the Flow-Field Pattern

3.2.2 The Flow in the Symmetry Plane in Front of and Behind the Cylinder

3.2.3 Rise and Evolution of the Global Separation Area

3.2.4 Evolution of Local and Integral Aerodynamic Characteristics

3.3 Flow over a Frontal Surface of a Circular Cylinder

3.3.1 Forward Stagnation Point

3.3.2 Frontal Surface of a Cylinder

3.4 The Flow in the Afterbody

3.4.1 Nonuniqueness of the Problem Solution

3.4.3 Characteristics of the Separation Area

3.5 Local Aerodynamic Characteristics

3.6 Integral Aerodynamic Characteristics

Chapter 4: Elliptic Cylinder in Supersonic Perfect Gas Flow

4.1 Simulation Conditions

4.3 Aerodynamic Characteristics of Elliptic Cylinder

Chapter 5: Supersonic Viscous Gas Flow Over a Sphere

5.1 Verification of the Numerical Simulation Approach

5.1.1 Simulation Conditions

5.1.3 Local Characteristics of the Sphere

5.1.4 Integral Characteristics of the Sphere

5.2 Instantaneous Launch with Supersonic Velocity

5.2.1 Evolution of the Flow Field

5.2.2 Distribution of Gasdynamic Variables along the Flow Symmetry Axis

5.2.3 Rise and Evolution of the Global Separated Flow Area

5.2.4 Evolution of Local and Integral Aerodynamic Characteristics

5.3 The Effect of Reynolds Number of the Flow-Field Pattern and Aerodynamic Characteristics

5.3.2 Local Aerodynamic Characteristics

5.3.3 Integral Aerodynamic Characteristics

Chapter 6: Two-Dimensional Axisymmetric Bodies with a Narrow Groove on the Frontal Surface in Supersonic and Hypersonic Flow

6.3 Flat Plate with a Narrow Groove in Supersonic Flow

6.3.1 Flow-Field Pattern Inside the Groove

6.3.2 Pressure Distribution near the Groove and Along Its Walls

6.3.3 Recovery Temperature (Enthalpy) of the Plate and the Groove

6.3.4 Heat Transfer Coefficient on the Flat Plate

6.3.5 Heat Transfer Simulation on the Groove′s Walls

6.3.6 Analysis of the Similarity Laws

6.4 Model of the Martian Probe “Pathfinder” with a Narrow Groove in Hypersonic Flow

6.4.1 Verification of the Numerical Model

6.4.2 Recovery Temperature of the Blunt Body with a Groove on Its Frontal Surface

6.4.3 Heat Transfer on the Isothermal Surface

6.4.4 Temperature Conditions on the Frontal Surface under the Full-Scale Flight Conditions

6.5 Model of the MSRO Probe with a Narrow Groove in Hypersonic Flow

6.5.1 Local Aerodynamic Characteristics on the Frontal Surface of the Probe

6.5.2 Recovery Temperature of a Narrow Groove

6.5.3 Heat Transfer on the Isothermal Walls of the Groove

Chapter 7: Blunt Axisymmetric Bodies in Supersonic and Hypersonic Flow at Zero Angle of Attack

7.1 Hypersonic Flow over the Model of Mars Pathfinder

7.1.1 Simulation Conditions

7.1.3 Aerodynamic Characteristics

7.2 Hypersonic Flow over the MSRO Probe Model

7.2.1 Experimental and Simulation Conditions

7.2.2 Flow Field in the Test Chamber of the Wind Tunnel

7.2.3 General Flow-Field Pattern and Local Aerodynamic Characteristics of the Model

7.2.4 Verification of the Numerical Simulation

Section 2. Numerical Simulation of Two-Dimensional Problems of External Aerodynamics

Chapter 8: Mathematical Problem Statement and Numerical Analysis

8.1.1 Differential Navier–Stokes Equations

8.1.2 Boundary and Initial Conditions

8.1.3 Differential Reynolds Equations

8.1.4 Boundary and Initial Conditions

8.2 Approximation of Equations

8.3 Solution of Nonlinear Discretized Equations

Chapter 9: Verification of the Numerical Simulation Method

9.1 Sharp Circular Cone with Half-Angle θ<sub>c</sub> = 15° at M<sub>∞</sub> = 10.4

9.1.1 Simulation Conditions

9.1.2 Local Aerodynamic Characteristics

9.2 Sharp Circular Cone with Half-Angle θ<sub>c</sub> = 4° at M<sub>∞</sub>

9.2.1 Simulation Conditions

9.2.2 Experimental Conditions

9.2.3 Local Aerodynamic Characteristics

9.2.4 Integral Aerodynamic Characteristics

9.3 Class of Sharp Elliptic Cones

9.3.1 Experimental Conditions

9.3.2 Simulation Conditions

9.3.3 Zero Angle of Attack

9.3.4 Nonzero Angle of Attack

Chapter 10: Sharp Circular Cone in Supersonic Perfect Gas Flow

10.1 Simulation Conditions

10.1.1 Mach Number M∞ = 4

10.1.2 Mach Number M∞ = 5

10.1.3 Other Mach Numbers

10.2 Thin Sharp Circular Cone at M<sub>∞</sub> = 4

10.2.1 Flow Field Pattern

10.2.2 Local Aerodynamic Characteristics

10.2.3 Integral Aerodynamic Characteristics

10.3 Thin Sharp Circular Cone at M<sub>∞</sub> = 5

10.3.1 Flow-Field Pattern

10.3.2 Local Aerodynamic Characteristics

10.3.3 Integral Aerodynamic Characteristics

10.4 Effect of Reynolds Number at M<sub>∞</sub> = 5 and α = 8°

10.4.1 Flow-Field Pattern

10.4.2 Local Aerodynamic Characteristics

10.4.3 Integral Aerodynamic Characteristics

10.5 Effect of Mach Number

10.5.1 Flow-Field Pattern

10.5.2 Local and Integral Aerodynamic Characteristics

Chapter 11: Thin, Sharp, Elliptic Cone in Supersonic Viscous Perfect Gas Flow

11.1 Simulation Conditions

11.2 Classification of the Flow Modes

11.4 Local Aerodynamic Characteristics

11.4.1 Zero Angle of Attack

11.4.2 Nonzero Angle of Attack

11.5 Integral Aerodynamic Characteristics

11.5.2 Effect of Reynolds Number

11.5.3 Effect of Mach Number

11.6 Effect of the Angle of Attack and of Sideslip Angle

11.6.1 Numerical Simulation Conditions

11.6.2 Flow-Field Pattern

11.6.3 Aerodynamic Characteristics of the Elliptic Cone

Chapter 12: Blunt Axisymmetric Bodies at an Angle of Attack in Supersonic and Hypersonic Flow

12.1 The Model of Mars Pathfinder at M<sub>∞</sub> = 6

12.1.1 Numerical Simulation Parameters

12.1.2 Flow Field Pattern

12.1.3 Local Aerodynamic Characteristics

12.2 The Model of Mars Pathfinder at M<sub>∞</sub> = 19.8

12.2.1 Numerical Simulation Parameters

12.2.2 Flow-Field Pattern

12.2.3 Local Aerodynamic Characteristics