Gas flow calculation method of a ramjet engine

Gas flow calculation method of a ramjet engine

Kirill Kostyushin, Anuar Kagenov, Ivan Eremin, Konstantin Zhiltsov, and Vladimir Shuvarikov

Citation: AIP Conference Proceedings 1899, 060012 (2017);

View online: https://doi.org/10.1063/1.5009883

View Table of Contents: http://aip.scitation.org/toc/apc/1899/1 Published by the American Institute of Physics

Gas Flow Calculation Method of a Ramjet Engine

Kirill Kostyushin

^a)

, Anuar Kagenov

^b)

, Ivan Eremin

^c)

, Konstantin Zhiltsov

^d)

and Vladimir Shuvarikov

^e)

National Research Tomsk State University, 36 Lenina Avenue, Tomsk 634050 Russian Federation

a)Corresponding author: kostushink@niipmm.tsu.ru

b)anuar@ftf.tsu.ru

c)iveremin@niipmm.tsu.ru

d)konstantin@niipmm.tsu.ru

e)shuvarikov@niipmm.tsu.ru

Abstract. At the present study calculation methodology of gas dynamics equations in ramjet engine is presented. The algorithm is based on Godunov`s scheme. For realization of calculation algorithm, the system of data storage is offered, the system does not depend on mesh topology, and it allows using the computational meshes with arbitrary number of cell faces. The algorithm of building a block-structured grid is given. Calculation algorithm in the software package

“FlashFlow” is implemented. Software package is verified on the calculations of simple configurations of air intakes and scramjet models.

INTRODUCTION

Nowadays different types of power devices of rocket technology are worked out. One of such scheme is ramjet.

In the paper [1] the numerical simulation of the performance of scramjet engine model with pylon set located in the inlet is done. In investigation [2], the numerical analysis of the gas flow in the air intakes of the scramjet engine is carried out. In work [3] the calculation of the gas flow in the combustion chamber is done. Because of complexity of construction and variety of physico-chemical processes, that are going on in ramjet, in order to receive local and integral characteristics it is necessary to conduct complex numerical calculations of gas flow in air intake, combustion chamber, nozzle and jet [4, 5]. At the conduction of such calculations, topology and size of the mesh play an important role. In the simple areas can be used one-block structured grid. However, to solve the gas dynamics equations of in complex areas, typical of ramjet it is necessary to use either unstructured grids or multi- block structured grids [6]. When using block-structured grids the computational domain is divided into simple subdomains. In each subdomain, a structured grid is built and then the coupling of computational meshes between sub domains is set. General weak point of the meshes, which are used at the calculations in complex areas is the dependence of solver algorithm on mesh topology. In this case at the little changes of mesh topology (alteration of face number in the cell, sub domain form and position) it is necessary for the solver algorithm modification. In this regard, it is vital to develop the universal algorithm which will allow to conducted calculations using the meshes with arbitrary number of cell faces.

The aim of this paper is to develop a methodology for calculations of the gas flow in gas-dynamic tracts of a ramjet, which consists of air intake, a combustion chamber, nozzle and supersonic jet.

CALCULATION METHODOLOGY

The Euler equations, which govern the compressible, inviscid gas flow in three-dimensions are [7]:

_³³³

_G ^dG

_³

_S ^{˜d 0}

dt

d

U U

v S (1)

_³³³

_G ^dG

_³

_S

^pˆ

^{˜d 0}

dt

d

U

v

U

vv I S (2)

_³³³

_G^edG

_³

_S

^{e p}

^{˜d 0}

dt

d v S (3)

where:

U

– density;

t

– time;

p

– pressure; ^v

>

^u,v,w

@

^T – gas velocity;

I ˆ

– 3×3 identity tensor; ε – specific internal energy; ^e

^UH

^U

^{u2 v2 w2}

² – total energy per unit volume;

G

– final volume in three- dimensional space;

dG dxdydz

– element of volume;

S

– surface limiting the volume

G

;

d S n dS

– orientable surface element

S

, where n – external normal unit vector to the surface

S

;

dS

– differential surface element.

System (1) – (3) is closed by equation of state:

U H k 1

p

(4)

In order to build difference scheme the finite volume method is applied. Therefore, we will cover all computational domain with discrete cells consisting of arbitrary convex polyhedrons with volume

G

_i, where

,...

2 , 1

i

– volume number, and m(i) – face number of

i

finite volume. Each face of

i

finite volume has surface Sj, where j 1,2,...,m(i). Approximation of integral equations in each of polyhedrons obtains in the following way:

⁰

1 1

˜ '

¦

mi

j

j j j k

i k i

i

R

G U t U V S

(5)

⁰

1 1

1

˜ '

¦ ¦

mi

j j j i

m j

j j j j k

i k

i

i R P

G

U

v t

U

v V V S S

(6)

⁰

1 1

˜

'

¦

mi

j

j j j j k

i k i

i

E P

t e

G e V S

(7)

As a result we will get Godunov`s scheme [8] for custom computational mesh. HereS_j n_jS_j, and

' t

– time step. Lower index

i

signifies value of functions, which are referred to mass center of

i

polyhedrons, and lower index

j

signifies values referred to center of

j

face of discrete cell. Upper index

k

signifies the number of time step. “Large” values of R,

V

,

P

,

E

signify correspondingly density, speed, pressure and total energy at faces of control volume. These values are defined from the solution of Riemann problem [8] in the direction of outward normal. Also for the flow calculations can be used other methods, based on approximate solutions of Riemann problem: Roe, Osher, HLL, HLLC, HLLE and others [9]. Scheme (5) – (7) is easily generalized for two dimensional and axisymmetric approaches [8].

System of Data Storage and Solver Algorithm

The simplest approach for organization the solver algorithm is the approach based on data storage in isolated arrays and their further correlation by using indexes of corresponding elements. This approach quite quickly allows developing solvers for meshes with topology known in advance, however, even at the slightest alteration of mesh topology it is necessary to modify the data storage system and solver algorithm. In this regard, for realization of solver algorithm, adapted to arbitrary face number in cells, it is suggested the following data structure should be used. (Fig. 1):

1. Class «Cell». Instances of class «Cell» keep information about cell geometry (mass center, area and so on) and gas parameters in cell.

2. Class «Face». Instances of class «Face» keep information about geometry (area or length, normal vector) and gas parameters (big values) at the cell face.

3. Data model of computational mesh is presented in the form of collections of class instances «Cell» and

«Face»

4. Interconnection «Cell» - «Face» is realized by storage of collection references on class instances «Face» in class «Cell», and connection «Face» – «Cell», by storage in class «Face» references on class instances

«Cell».

FIGURE 1. Interconnection between classes «Cell» and «Face»

Such system of data storage does not have any dependence on computational mesh topology, and allows conducting calculations on meshes of custom configuration [10]. When using object-oriented programming languages, scheme of solver (5) – (7) can be written in a quite simple and compact way (Fig. 2).

FIGURE 2. A listing of the example of solver source code of time step. Language C#.

Here «cells» are collection of class instance cell, «cell.faces» are collection of reference on cell faces,

«stream_1», «stream_2», «stream_3» – sum of flux through faces of computational cell for conservation equation of mass, momentum and energy correspondingly. Fields «face.Gas.Ro», «face.Gas.V», «face.Gas.P» signify density,

velocity and pressure at the face of computational cell and calculate by solving Riemann problem (not given in the listing). «face.S», «face.N» – area and unit vector of outward normal. «cell.Gas», «cell.Gas_Next» are gas parameters in the сell center at the current and next time step correspondingly, «cell.G» is cell volume, «dt» is current time step, which calculated from Courant–Friedrichs–Lewy condition. Fields «V», «N» and variable

«stream_2» – instances of class «Vector» for which standard vector operations are identified.

The use of suggested data storage system and solver algorithm allow conducting calculations at meshes with arbitrary topology.

Mesh Coupling

At the usage of blocked structured or unstructured grids, a task of meshes coupling among sub domains comes up. In operation, for meshes coupling among sub domains the algorithm, which is based on edges division of the cells (on the coupling line) into composite edges, is used. As an example, let look at the coupling algorithm of two two-dimensional computational sub domains, shown on Fig. 3.

1. For each boundary cell (c) from the first sub domain, the edge (p2-p4) is given out, which join on the boundary of sub domains separation.

2. Boundary cells of the second sub domain (c2, c3) are defined, the ones that have common boundaries with the analyzed cell (c).

3. The vertex (p3) of cells (c2, c3) is defined, which situated on the edge (p2-p4).

4. The edge (p2-p4) of the cell (c) is broken into several edges (e2, e3) along the vertex (p3).

5. Algorithm is repeated for each cell from the first domain.

FIGURE 3. Coupling scheme of two computational sub domains

For more complex configurations of couplings the algorithm is analogous. Algorithm can be used for coupling of sub domains with different topology, and also it is easily generalized on three-dimensional computational meshes.

CALCULATION OF SUPERSONIC FLOW IN AIR INTAKE

To verify algorithm and methodology of calculation a series of calculations on a simple flat hypersonic air intake was conducted. General scheme of air intakes under consideration are given on Fig. 4.

FIGURE 4. Air intake scheme

Where

H 1

– characteristic dimension;

L 7

– length of air intake;

T 10 q

– inclination angle at inlet of air intake,

x

_e – coordinate, defining position of air intake’s upper side, x_g – position of corner point, h_g – height of straight section of air intake. Design position of air intake corresponds to Mach number

M

₀

5 . 3

. At this mode, in the inviscid gas flow, the shock wave makes with x-axis the angle

T

_w

18 . 67 q

[11]. Therefore, the position of air intake’s upper side is defined by:

96 .

w

2

e

ctg

x T

. (8)

Calculations for three configurations of air intake have been conducted (Table 1). To build the computational mesh the computational domain was divided into 4 computational sub domains, which are shown on Fig. 5.

ТABLE 1. Geometric parameters of air intake.

hg

x

_e xg

0.2 2.96 4.54

0.3 2.96 3.97

0.5 2.96 2.83

FIGURE 5. Computational domains for air intake’s configuration with h_g 0.2

At all inner elements of air intake’s construction, the boundary condition is solid wall. At exit section of air intake – supersonic outflow. At other boundaries of outer domains – supersonic inflow with the speed

M

₀. Working gas is air. At Figure 6 distribution of Mach number is shown for three configurations of air intakes. For straight section height h_g 0.5 shock wave from upper side of air intake gets into straight section of air intake area. For h_g 0.3 shock wave from upper side of air intake gets into straight section of air intake at the neighborhood of corner point. For h_g 0.2 shock wave from upper side of air intake gets into wedge surface, and disturbances spread up in the flow for large distances.

FIGURE 6. Distribution of Mach number for air intake with straight section height h_g 0.2 ^(a), h_g 0.3 ^{(b) и} h_g 0.5^(с) Qualitative flow picture corresponds to solutions, that were received in [11] for viscid gas flow at large Reynolds numbers Re=10⁶.

FLOW CALCULATION IN SCRAMJET ENGINE

For the test geometry was chosen classical scheme of ramjet. This scheme consists of air intake with center body, combustion chamber and nozzle (Fig. 7).

FIGURE 7. Geometry of model ramjet. Sectional view

To build multi-block structured grid, the computational domain (including the scramjet and outer domain) was divided into rectangular subdomains (Fig. 8).

FIGURE 8. Decomposition of computational domain

Structured computational grid was built in each subdomain. A computational mesh coupling among sub domains was conducted with the use of above-mentioned algorithm. As an example, Fig. 9 shows a part of computational mesh in area A from Fig. 8.

FIGURE 9. Computational mesh in area A

Boundary condition at inner and outer engine elements was for solid wall. On left boundary of outer computational domain - supersonic inflow (pressure: 0.1 atm., temperature: 216 K, speed: 6 Мach). At other outer boundaries – supersonic outflow. The task was solved, with starting distribution in all sub domains equal to parameters of supersonic inflow. Working fluid is air. Developed software package «FlashFlow» [10] was used for building geometry and computational meshes and also for conducting calculations and results visualization.

At Figure 10 distribution of Mach number is shown, where can be seen that supersonic stream is decelerating at inlet of air intake. Gas flow in combustion chamber is mainly supersonic, with local subsonic zones. Qualitative stream picture corresponds to familiar solutions at cold purging of gas dynamic routes of ramjet [12].

FIGURE 10. Distribution of Mach number for scramjet engine

CONLUSION

A methodology for calculating of the gas dynamics equations in gas-dynamic tracts of a ramjet based on the Godunov scheme is developed. The calculation algorithm and the data storage system witch independent of the mesh topology for to use it with an arbitrary number of cell faces are developed. An algorithm for constructing a block-structured grid is given. The calculation algorithm is implemented as a software package. The calculations of gas flow in the air intake and in the scramjet engine are done. The results of the test calculations are in good agreement with the experimental data, which obtained by other authors [11, 12].

ACKNOWLEDGMENTS

The work executed at financial support of the Ministry of education and science of the Russian Federation in the framework of the state assignment, project No 9.9063.2017/8.9.

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