Numerical study of viscous gas flow in the wake of a plane body

NUMERICAL STUDY OF VISCOUSGAS FLOW IN THE WAKE OF A PLANE BODY* V. I. MYSHENKOV Moscow (Received

GAS flow in the wake behind Navier-Stokes velocities

equations,

17 May 1971)

a plate

of finite

at subsonic,

the Navier-Stokes The problem

step difference numbers, nature

equations,

a plate of finite at subsonic,

is solved

scheme

of the Lax-Wendroff

the temperature

factor,

the

flow

paper is an extension

and development

The computational

results

type.

explicit

The influence

two-

of the Re and M

of the open viscous

layer and the The appear-

in the flow are investigated.

agree qualitatively

with the available

The

theoretical

of the problem

the gas flow in the wake behind that the flow parameters

a plate

are specified

coordinate

from the bottom edge of the plate.

system)

of finite at infinity

The solution

in the upper quadrant

facilitate

as in 111, is mapped by the mapping rn . t o a rectangular

region

Mat. mat.Fiz.,

12, 3, 673485,

108

1972.

with respect

of the plane, 5 =

with coordinates ~-

under to

6, up-

will be sought by means

equations

integration,

thickness

and at some distance

of the Navier-Stokes

*Zh. urchisl.

using

of [l].

Formulation

t and 17(5, q is a rectangular

Y =11/1(Wf~) O
and small supersonic

data. 1.

the condition

is considered

in the wake, is considered.

of bottom separation

and experimental

method,

the thickness

of the gas, on the flow parameters

Consider

thickness

near-sonic

by the establishment

ance and development

stream

using

supersonic

(0.3 \< M ,< 2, Re = 100) and Pr = 0.71 and 1 ,< Re,< 1000 (at M =

flow velocities 0.3).

is considered

and small

(0.3 ,< M < 2, Re = 100) and Pr = 0.71 and 1 ,( Re < 1000 (at M = 0.3).

The gas flow in the wake behind using

thickness

near-sonic,

which,

to

E /1/( E” + 1)) 0 <

r <

1,

Numerical

study of viscous

109

gas flow

0.8 I R6 t

0

012

0.V

I

6

0.6

0.8

.T

FIG. 1.

The,non-stationary

equations

in the dimensionless and thermal

of motion and energy,

form, and the temperature

conductivity,

the equation

dependences

will be used in exactly

of state

of the viscosity

the same form as (1.2)-(1.4)

of [ll. The problem solution

exists

will be solved and is unique

by the establishment

under reasonably

method,

smooth boundary

assuming

that the

conditions.

A

left to right flow is assumed. The boundary (1)

p =

(2)

p = f&j,

(3)

u =

u =

y =

yu, 0 d

(4)

u =

conditions

i$ I dy =

1, L; =

are 0, e = e,

u = b(y), 0, e = 5 <

e,,

u = f3h),

s=1,o~Yyl;Y=1,o~

e =

f&(y) for t = 0, y. 5s y <

p is given by the equation

X0, 2 =

3u / ay =

for

X0, 0 <

de / dy =

of continuity

for

y < yll;

v = 0 for

Y=

0. 50 d

5 <

1.

1;

V. I. Myshenkov

110

Here, fi, i = 1, 2, 3, 4, are arbitrary the existence viscous

wall layer.

fi were constant,

For simplicity,

subscripts infinity).

of the plate

x and y velocity w and m denote

satisfying

it was assumed

equal to the corresponding

y0 are the x, y coordinates the density,

functions,

the conditions

e.g. the p, u, v and e distribution

of the solution,

surface

components, parameters

in the

in most computations

parameters (Fig.

for

functions

in the entrant

that the

flow;

x,,

l), and p, u, LI and e are

and internal

energy

respectively.

at the wall and in the entrant

The

flow (at

!t was previously mentioned in [l] that the flow symmetrization condition somewhat stabilizes the flow and is unjustified at large Re numbers, for which the flow becomes non-stationary (formation of Karman streets). Another point which remains valid is that regarding the statement of the boundary value of the density

when x = 1, O,< y< 1 111,

InitiaE conditions. in the computational however,

The initial region

to accelerate

values

the computation,

Re, Pr and M numbers,

of the field of variables

were originally

as initial

specified when solving

a problem

data we used the solution

The solution

will be regarded

the longitudinal the condition

as steady-state

velocity

components

(Here,

Later,

for specific

previously

for certain other vaiues of the characteristic parameters. are the Reynolds, Prandtl and Mach numbers.)

between satisfies

p, u, u, e

quite arbitrarily.

obtained

Re, Pr and M

if the norm of the difference

throughout

the computational

region

(I.11 Here, uk is the value the numerical

of u at the instant

The problem

will be solved

scheme

[2], approximating

second

order of accuracy

in detail

t = tilt, where At is the time step in

computation. by means

the initial O(h’).

of a Lax-Wendroff

system

of Navier-Stokes

The structure

of the difference

two-step

difference

equations

to the

scheme

is given

in fll.

Missing

fictitious

(at the plate surface) third degree.

values

of the variables

will be defined

by means

u, u and e at the region of polynomials

boundaries

of at least

the

2. Aspects of the method In order to select

the difference

scheme,

the problem

was initially

investi-

study

Numerical

gated by means differences)

of three explicit

with a staggered

Wendroff second-order of these

schemes

variety

The problem described,

schemes.

(Notice

was solved

111

scheme

that some previously

and internal

(one-sided

and Brailovskaya

been discovered,

of external

with an entrant

a first-order

schemes:

point distribution,

have lately

of problems

of viscous gas flow

[3] and Lax-

unknown

features

and have been used to solve

a

flow [4, 51.

by the establishment

air flow velocity

method in the statement

U = 100 m/set,

above

and temperatures to a Mach num-

0.71, which correspond

T,=T,=3OO”K,Re=lOO,Pr= ber M = 0.288. The computations similar

solutions,

wake.

On continuing

showed

involving

that,

at the initial

formation

ttu computation,

tion, the Brailovskaya values of the density

however,

to the appearance

values

of the mesh.

Similar in the recent densities vative

points

results

all three schemes

region

give

in the immediate

up to establishment

of the solu-

and ‘staggered” schemes give continuously decreasing on the rear wall in the neighbourhood of the corner. In

the long run, this leads at certain

stage,

of a back-flow

at the rear corner

paper [4].

of negative

were obtained

Here, the authors

in the computation

by using

and temperature

for the “staggered”

avoided

special

density

the appearance

boundary

scheme

of negative

conditions

in the conser-

form.

In [S], where problems scheme,

no such effects

M b2; action

of flow past blunt bodies

were observed

this can be explained of the smoothing

It may therefore operate giving becomes

badly

by insufficient

time and the fairly

that the “staggered”

with relatively

decreasing

strong

and Brailovskaya

weak rarefaction

density

values

“waves”

which are too low.

schemes

(M = 0.288), This effect

much more marked as M increases.

It is only the Lax-Wendroff creasing

computing

by Hrailovskaya’s

even at numbers

operator.

be concluded

in regions

monotonically

were solved

in the bottom region

density

to be unique

scheme

in the neighbourhood

when computing

that does not give monotonically

of the corner.

with different

initial

The solution

conditions

in the range 14 Re< 100. These results proved a decisive of the Lax-Wendroff scheme for solving our problem. Meshes

of the following

0.05, AX = 0.025 and

types

were used: Ax = hy = 0.05,

AX =

de-

was found

and Re numbers argument

in favour

Ay = 0.1, Ax = by =

hy = 0.025.

The time step was

V. 1. Myshenkov

112

found experimentally. obtained llatory,

With a mesh step Ax = Ay = 0.1, a stationary though the p, u, u and e distributions

quite quickly, especially

existence

with respect

of a saw-tooth

It is interesting

to the x coordinate;

solution

close to the limit of scheme stability, be observed; as At decreases, these At was always

this is explained

in second-order

to note in this context

solution

are strongly

schemes

that,

is

osci-

by the

[61.

given certain

values

of At

time oscillation of the variables may also oscillations disappear. In our computations,

below the limit of stability.

As the mesh steps Ax and Ay decrease, the amplitude of the oscillations drops considerably, while it increases somewhat as the Re number increases, especially

on the right-hand

is a maximum.

boundary,

The solution

time for mathematical

establishment

to the time for physical

accuracy

of the coordinates

substantially.

of the solution

establishment

We thus come up against tational

where the compression

time then increases

However,

corresponds

the

quite closely

of the flow process.

the contradictory

requirements

of improved

compu-

computing time, and the decision regarding these Flow computations with Re = 100 and M = 0.3 has to be based on a compromise. with different mesh steps showed that the solutions are convergent as Ax and Ay decrease.

and reduced

The divergences

between

the main flow parameters

in the wake

(pressure and longitudinal velocity) for the cases Ax = Ay = 0.05 and Ax = Ay = 0.025 amounted to 2-3%, i.e. to economize on time, the former, coarser mesh could be used for most computations. Due to the long computing with a step of less than 0.025 were not used. The smooth distribution by averaging determining

the final

of the variables

computational

the flow parameters

the right-hand A certain

boundary;

divergence

explained

by the insufficient

objective

assessment

Solution

time and the absence

“wave.”

Monotonically decreasing in the neighbourhood of the corner into the range of large M without

badly

greater

of

at the point x = 1.

data from those

of establishment

operates

The maximum error in

by the singularity

with Mach numbers

scheme

paper was obtained

with large Re in the neighbourhood

of the present

computing

of the instant

of the problem

even the Lax-Wendroff

data over x and y.

occurred

this is explained

quantitative

in the present

time, meshes

of [ll is

of a satisfactorily

in [l]. than 0.3 reveals

in the region

that

of the rarefaction

density values at the bottom wall appear here at M = 0.8, and it is impossible to move further modernizing

the scheme.

Numerical study of viscous The smoothing

operator

the same way as in

is introduced

gas flow

113

into the scheme

at each time step in

[sI:

(Fm,n)o = Fm,,+ e -& AF,,, and enables

the problem

to be solved

with M as high as M = 2.

vector F = (p, u, II, e), AF: R is the difference and 0 ,’ c< 0.2. Check computations

performed

analogue

with different

Here, F is the

of the Laplace

smoothing

coefficients

operator,

0 ,< t \<

0.5 with M = 0.8 and 1 showed an insignificant

change in the flow parameters for with M = 0.8 the main flow parameters in the wake (u and

small 6. For instance,

p) only differed by 2-4% as between 6 = 0 and t = 0.1. This enables smoothing operators to be used to obtain solutions with large M, though this device cannot be regarded coefficients

as a genuine answer to the problem. The minimum possible smoothing were employed: t = 0.1 and M = 0.8 and 1, and t = 0.2 with M = 2.

The introduction t = 0.1-0.2

into the computational

is in essence

equivalent

a term of higher order of smallness bound to have a slight The greatest

energy.

influence

scheme

to adding

of a smoothing

to the initial

than the dissipative on the solutions

differential

terms.

of the equations

error may occur when solving

operator

the equation

This

with

equations is naturally

of motion and of continuity,

due to the absence of dissipative terms, and notably, in those regions where the viscous forces are fairly high. In this case the over-all error in solving the difference

system

of Navier-Stokes

equations

also increases.

3. Results Wake flows ere investigated types

of gas, and the influence

T lo, m = condition

for different

Re and M numbers

on the flow parameters

and different

of the temperature

factor

and the thickness of the viscous layer (left-hand boundary TWIT, fi). Perfect gases were investigated, most of the computations being

performed for air. As the characteristic magnitudes of the problem were taken the entrant flow parameters and the length I,, proportional to half the thickness of the rectangle ;: L = 1.733 1; Pr = 0.71. Solutions

were obtained

for the following

values

of the characteristic

para-

meters:

1, T, =3OO”K, 1
Tw,

m

=

V. 1. Myshenkov

114

(2) Tw... = 1,T, = 300"K,Re = 100,0.3< M < 2, Ax= Ay = when investigating the influence con&, air, helium and propane: 0.05, fi = of the M number and type of gas; (3)

0.6
0.05, fi =

<

2,T, = 300"K, Re = 100,M = 0.3, Ax =Ay =

con&, air:

when investigating

the influence

of the temperature

factor;

K, Re = 100, M = 0.3, Ax= Ay = (4) T,, m = 1, T, = 300" air: when investigating the influence of the left-hand 0.05, fi = variable, boundary

condition

The thermal Dependence throughout ‘1

Yt

(the thickness and physical

of the transport that Sutherland’s

6 of the viscous

parameters

layer).

of the gases

factors

on pressure

formula

applied.

was ignored:

=2 0.6

were taken

from [7, 81.

it was assumed

*=2 0.6

-

0.1

u=0.3

@iz$ -0.08

;c,

F

=2

0.6

-

7

FIG. 2.

1. Influence of the Re number. The computations

showed that the wake flow behind a plate with Reynolds numbers Re ,( 1.7 and M = 0.3 is of the unbroken type. With Re = 1.7 quite a large low-velocity region appears round the rear critical point; this transforms, as the Re number increases, into a reverse

sludy of viscous

Numerical

gas flow

115

M

FIG. 3.

U6-

lg Re FIG. 4. flow region,

causing

the flow to break up at the bottom.

The viscous

then such that the flow near the sharp edge turns through a break; the break (the start of the separating streamline

forces

are

a right angle without I/I = 0) occurs at the

bottom of the plate (and not at the corner). The possible predicted Stokes’

existence

in 191, on the basis

equations

experimentally

of a break in the flow below the corner of a qualitative

for the neighbourhood

at supersonic

failed to be obtained, enough.

because

speeds

study of the analytic

of a singular

in [lp,

111; here,

the experimental

point,

was earlier solution

of

and was investigated

a convincing

confirmation

data were not analyzed

exactly

As the Re number increases further the size of the backward flow region increases, and the flow break shifts upwards relative to the plate bottom, towards

V. 1. Myshenkov

116

0

0

1

0.2

0.4

1 u

b

FIG. 5. the corner (see Figs. Re of the coordinates

2, a and 3, a, showing the streamlines y*, x* of the break point at M = 0.3).

and the variation with The break coordinate

y*, measured along the bottom wall, was found by extrapolating the stream function Ifi and longitudinal velocity component u; when $ and u are small, such extrapolation

can lead to a serious

error in determining

ye.

Similar results regarding break displacement as Re varies (for a developed break) were obtained in the recent paper [4], on the basis of a numerical solution of the Navier-Stokes The instant

equations,

primarily

when the break appears

of the bottom pressure

p, (the pressure

then sharply

(see Fig. 4, a).

increases

for an incompressible is most clearly

at the centre

revealed

of the plate

The determination

fluid. by the behaviour bottom),

of the instant

which of break-

ing from the change of sign of the longitudinal velocity component is less accurate, due to the low velocities in the wall zone, which are at this instant comparabie with the computational error. In the long run, this leads to the determination of a later instant of breaking (relative to the Re number), see Fig. 4, a. Figure 5, a shows the variation of the longitudinal velocity component in the plane of flow symmetry at M = 0.3 with different Re numbers. It can be seen from the curves that, as Re increases, so do the magnitudes of the backward velocities as well as the size of the backward flow region.

Numerical study of viscous gas flora It can be seen from Fig. 5, b that the pressure most strongly

at low Re numbers,

number increases smooths

flow region

out and at Re = 100 becomes

corner that a significant

at the plate bottom varies

when the flow is still

and a backward pressure

117

unbroken.

As the Re

forms, the pressure

almost

constant.

drop occurs,

distributions

It is only at the plate

due to the rarefaction

“wave.”

As the Re number increases the pressure at the centre of the plate bottom (the bottom pressure) remains virtually unchanged with the instant of breaking, after which it rises

sharply

(Fig.

4, a).

up to Re = 1000, the bottom pressure qualitative

agreement

non-stationary.

in the norm of the difference

during the computation. component property

obtained

it.

The values

at different

But in spite

was detected. amplitude

perceptible

is explained

with completely

process

2. Influence

of the backward

flow regions.

The influence

2, b and 3, b).

region

meantime

A further

The relative

changes increase

somewhat

Some

Re numbers

of the M number was investi-

The computations

showed

u distribution

that, as

and narrows

in the dissipative

while the maxima of u remain virtually

to M 3 I causes

tion of the back flow region {Figs.

5, a).

flow region elongates

velocity

has a small

smoothly as a function p, at infinity, with a

flow region (Fig.

M,< 2, at Re = 100.

flow

is given in Fig. 4, a.

with different

from 0.3 to 0.8, the backward

(Figs.

steady-state

time, no vortex break

curves

of the M number.

gated in the range 0.34 M increases

in the pressure sizes

velocity

the bottom pressure

value

minimum in the backward

by the different

is (1.1)

and longitudinal

the computation

and its time-averaged

behaviour

At

oscillation

u in expression

The pressure in the plane of flow symmetry varies of x, from the pressure p0 at the bottom to the pressure scarcely

in

fluid.

also throw light on the non-stationary region,

of prolonging

of oscillation

vectors

of the pressure

instants

During the computational

lack of monotonic

of the Re number, monotonically,

A periodic

between

of the flow in the dissipative

outside

increase

to increase

with the data of [12] for an incompressible

Re = 1000 the wake flow becomes observed

On further

continues

both a widthwise

and lengthwise

2, b and 3, b) and obviously

leads

fixed. contrac-

to break-

less flow at some M > 2 (with Re = 100). An increase the flow, delaying

of M (with M >, 1) thus proves the appearance

with M = 1 the flow remains 0.3 quite a developed point is that,

breakless

back-flow

if the computational

to have a stabilizing

of a break behind

effect

results

at Re = 10.

are processed within

on

For instance,

at Re = 10 or even 15, whereas

region was observed

meter MRe es , the bre ak will commence

the plate.

with M =

An interesting

as a function

of the para-

quite a narrow range of this para-

118

V. 1. Myshenkov

meter, namely

0.3 to 0.355,

for the range of M numbers

investigated

(here,

L = 0.

FIG. 6. From Fig. 4, b, as M increases, The variation contraction

of the dissipative

from the edge);

as mentioned

above,

bottom centre also affected

displacement

this displacement

influence

and the influence

at the plate

with M is here probably

region (downward

With Re > 100, this spurious be avoided,

the pressure

of this bottom pressure

of the break point

leads

to a drop in pressure.

of break zone contraction

of M on the bottom pressure

falls. by

can obviously in the Upure”

examined

form. With Re = 100, the pressure same trend as M increases change

occurs

tonic variation slightly

from monotonic (Fig.

5, b).

below the cotner,

The pressure

variation

distribution small

break-away pressure

(as compared zone,

With M $1,

variation

to substantial

a minimum pressure

region

the i.e. a

non-monoappears

with the data of f131.

with x in the wake has the same monotonic As M increases,

with the p(x) curves

apparently

~-distribution

pressure

in agreement

for all M in the range 0.3 < M 6 2. pressure

on the plate bottom reveals

as in the case M = 0.3 when Re decreases,

caused

M >/ 1, a region

is not monotonic

of increased

for other M < 1) forms behind

by the compression in the different

ally for large M, the pressure becomes virtually flow region even with M = 2 (Fig. 6, Re = 100).

tail wave.

the

While the

wake sections,

independent

character

especi-

of y in the back-

Numerical

3.

influence

study

of the type of gas.

used to investigate

this influence,

of viscous

Three

gases,

helium,

distributions

longitudinal

etc) remain the same,

velocity

contraction

of the back-flow

is most pronounced lowering

distributions

For instance,

in the series region

in the wake (i.e.

propane

in length

as the Mach number increases,

O-

-O.l-

-0.2-

-LX?-

-0.4-

I

0.2

0.4

with only slight

- - air - - helium,

and

quantitaa slight

This trend

at M = 2. There

is also a

below the bottom wall.

0’

^^

were

the pressure

and width is observed.

of the point at which the flow breaks

vJ.Y

air and propane,

at Re = 100 and with 0.3 < M < 2; it was

found that the main flow parameter tive variations.

119

gas flows

1

0.6 a

0.8

x

1

0.4

0.6 b

0.8

x

1

FIG. ‘i.

The bottom pressure

variations

with M for the different

gases

can be seen

in Fig. 4, b. The pressure like the longitudinal

on the wall and in the back-flow velocity

u, for propane

(curve

region

proves

1) than for air (2) or helium

influence of the temperature factor Fw m. The influence was investigated for Re = 100: M = 0.3, and T, T, o3 = T,/T, 4.

varying

the wall temperature

in the range 160-600°K,

to be greater,

of this factor = 300°K, by i.e. the temperature factor

(3).

V. 1. Myshenkov

120

varied

in the range 0.6 \< ?;,

The computations little

effect

certain

m < 2.

showed

that the temperature

on the flow parameters,

types

of variation

and notably,

do become

apparent.

factor variation

on the bottom pressure, though As the temperature factor increases,

the bottom pressure and the pressure in the dissipative region larly, we find the same variation of the backstream velocities, connection

of these

with the bottom pressure

The computations temperature

factor,

of the break-away The behaviour seems

revealed

probably

The variation

fall sIightly. Simirevealing a direct

7, a).

in the back flow region

of the closeness

with the

of the right-hand

boundary

point x = 1.

in viscosity,

as the temperature and hence

factor

a reduction

increases

in the local

region.

with Tw cDof the dimensionless

heat transfer

characteristic,

number Nu, where

4T,,m--3Tm.,-Tm--i,n _ 3) s7 (1 2Ax(l+(y- l)M2/2T,,,)T,

Nan,, = a dependence

bottom.

The slight

(Cu, 0) of small

quantities

5.

(Fig.

of the main flow parameters

number in the dissipative

the Nusselt

gives

because

zone to the singular

to be due to an increase

Reynolds

no change

has relatively

of the form

. at the centre of the plate

fall in Nu at FU m = 1 away from the general

is explained

Influence

NU - F$$5

by the influence

in the computational of the viscous

of the error in the solution

law

NU -;: f

during

division

formufa.

layer thickness

6.

The influence

of the viscous

layer thickness 6 on the left-hand boundary of the computational region fi was investigated at Re = 100, WI= 0.3 on viscous layers of the sinusoidal and breakaway types. The break-away profile was taken as U,,, = 0 for n 6 n, + 6 and = 1 for n > n, + 6, where 6 is the thickness of the viscous layer y = Ay, y. = uO,il n&y. The distributions of the remaining variables (p, u, e) for the cases fi were assumed to be identical with those used above. The computations revealed only a slight change of bottom pressure as the viscous layer thickness increased, even for the case of a break-away profile; this agrees with the results of [41. As 6 increases the pressure distribution on the wall smooths out and becomes constant. Variation of 6 has most effect on the velocity distribution in the back-flow region, especially near the right-hand boundary, towards which the centre of the continuously increasing vortex moves

Numerrcal

as 6 increases backward

(Fig.

7, b).

velocities

The

pressure

increases, (Fig.

meantime A picture

in Fig.

8.

on the lateral

when

a barrier

The

(near

the bottom

wall),

the

virtually

shows

which

becomes

of the vortex

constant

an increasing

with

more marked

centre.

variable

deviation

The

size

6

as x

as 6 increases of the back-flow

increases.

of the streamlines

Clearly,

region

remains

wall,

minimum,

in the region

also

even section

which

of the bottom

and a pressure

121

gas flou

as 6 increases.

distribution,

7, b), appears

region

In the near-wake

decrease

in the neighbourhood

stud? o/ viscous

with

such

surface

of height

for the break-away an exotic

of the plate.

y = Shy

boundary

profile

In practice,

is mounted

with 6 = 4 is seen

fi,

condition this

on the plate

type surface

break-away

occurs

of flow is realized in the cross-

x = 0. author

thanks

B. S. Kirnasov

for useful

discussions.

Translated

by D. E. Brown

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