MHD compressible turbulent boundary-layerflow with adverse pressure gradient

M. Xenos, S. Dimas, and N. Kafoussias, Patras, Greece

Received January 5, 2004Published online: May 13, 2005 � Springer-Verlag 2005

Summary. The effects of the magnetic field and localized suction on the steady turbulent compressible

boundary-layer flow with adverse pressure gradient are numerically studied. The magnetic field is constant

and applied transversely to the direction of the flow (global or local). The fluid flow is subjected to a

constant velocity of localized suction, and there is no heat transfer between the fluid and the plate

(adiabatic plate). The Reynolds-Averaged Boundary-Layer (RABL) equations and their boundary con-

ditions are transformed using the compressible Falkner-Skan transformation. The resulting coupled and

nonlinear system of PDEs is solved using the Keller’s box method. For the eddy-kinematic viscosity the

turbulent models of Cebeci-Smith and Baldwin-Lomax are employed. For the turbulent Prandtl number

the extended Kays-Crawford’s model is used. The flow is subjected to an adverse pressure gradient. The

obtained results show that the flow field can be controlled by the applied magnetic field as well as by

localized suction.

1 Introduction

Control of flow separation was first introduced by Prandtl along with his boundary-layer

theory. Since then, many passive and active techniques have been developed for the prevention

or delay of flow separation. Passive techniques have been currently employed via blown flaps

on the tip of the aircraft wings or leading edge extensions and strakes on the nose of the wings

(slats) or via vortex generators on various points on the wings [1].

On the other hand, suction/injection has very often been used as an active aerodynamic flow

control technique to prevent transition from laminar to turbulent flow as well as turbulent flow

separation. It seems that it is the most effective technique for boundary-layer control, since it is

widely used by many aeronautic manufacturers for the preservation of the laminar flow, for

total drag reduction, for delaying the contamination of the leading edge and for lift optimi-

zation [2].

In many studies suction through a surface has been used for turbulent boundary-layer

control. Wilkinson et al. developed a hybrid suction surface for turbulent flows [3]. Gad-El-Hak

and Blackwelder suggested an approach called selective suction [4]. Sokolov and Antonia

studied the response of the turbulent boundary layer under an intense wall suction [5]. Another

suggestion for boundary-layer control is to use localized suction, that is to apply continuous

suction in a region and not at the whole length of the boundary surface [6].

It has been found that injection has an inverse influence compared to suction [6]. By blowing

air, the skin friction in a turbulent boundary layer is reduced but the near wall turbulence

Acta Mechanica 177, 171–190 (2005)

DOI 10.1007/s00707-005-0221-7

Acta MechanicaPrinted in Austria

activity becomes more intense [7]. The combined influence of localized injection and localized

suction retains the boundary-layer flow, reducing skin friction [8], [9]. Another means of

boundary-layer control is by heating or cooling the wall [10].

If the fluid is electrically conducting and under the influence of a magnetic field, an additional

flow control can also be applied. The idea of controlling the boundary-layer flow of an elec-

trically conducting fluid by electromagnetic forces dates back to the 1960s. Rossow was the first

who studied the incompressible boundary-layer flow over a flat plate in the presence of a

uniform magnetic field applied normal to the plate [11]. Bleviss investigated Magnetohydro-

dynamic (MHD) effects on hypersonic Couette flow under the influence of an externally im-

posed uniform magnetic field, normal to the wall. For the case of a thermally insulated wall he

showed that a tremendous decrease in skin friction but a significant increase in total drag is

obtained [12].

Recently, the influence of a magnetic field on the flow field has attained new attention as a

control technique for turbulent boundary layers. The magnetic field delays transition from

laminar to turbulent flow and separation of the turbulent boundary layer. A tremendous

reduction of skin friction will result from transition delay, since turbulent skin friction in

general is orders of magnitude larger than the laminar one [13]. The delay of separation also

reduces skin friction, because the separation phenomenon entails large energy losses, due to the

reversal of the flow and the considerable thickening of the boundary layer [14].

In supersonic and hypersonic flows, the gas can become weakly ionized either by viscous

heating at high temperatures or by artificially generated plasma at low temperatures [15]. When

the temperature of the gas is low then the dependence of the electrical conductivity r on the

temperature T is minimized and the conductivity of the gas is almost zero. In order to obtain an

acceptable value for r (e.g., 1.0 mho/m), seeding of an ion in the flow field must take place [16].

Using the direct exhaust from a combustion process as aworking fluid, the dependence of r on the

temperature T stands, the electrical conductivity is not zero and no seeding is required. Finally,

using short duration, high repetition rate and high voltage pulses a cold supersonic gas can be

ionized [17].

The numerical investigation of the two-dimensional turbulent boundary layer compressible

flow, over a finite smooth and permeable flat surface, with an adverse pressure gradient and

heat and mass transfer was studied in [6]. It was found there that the continuous suction/

injection applied on the wall significantly influences the flow field and the separation point. Also

the effect of the localized suction/injection when it is applied near the nose of the flat plate was

examined. The above control techniques were examined for adiabatic, heating and cooling wall.

The problem of a compressible turbulent boundary layer, under the influence of an applied

magnetic field, is an important issue [18]–[21], that becomes more interesting if the effect of an

adverse pressure gradient –which very often appears in the flowfield – is taken into consideration.

The aim of this work is the numerical study of the MHD, compressible turbulent boundary-

layer flow over a permeable flat plate, in the presence of a strong adverse pressure gradient. The

magnetic field is considered either constant and applied to the whole length of the plate (global)

or applied to a length near the separation point (local). Also, in this study the localized suction,

applied to the region of the separation point, is examined. The boundary-layer flow is con-

sidered turbulent and two turbulent models are employed, e.g., the Cebeci-Smith and

Baldwin-Lomax. The electrical conductivity of the fluid is considered either constant (low

initial temperature) or varying with the temperature (high initial temperature). From the

extensive analysis of the obtained results it is concluded that the magnetic field (local or global)

and the localized suction influence on the flow field and at the separation point render the above

applications as flow control techniques.

172 M. Xenos et al.

2 Mathematical formulation

We consider the steady two-dimensional adiabatic compressible MHD turbulent boundary-

layer flow over a smooth flat permeable surface. In a Cartesian coordinate system the surface is

located at

y ¼ 0; 0 � x � L; �1 < z < þ1;

and is parallel to the free-stream of a heat-conducting perfect gas flowing with velocity u1 in

the positive x-direction (Fig. 1). The fluid is assumed to be Newtonian, electrically conducting

and the plate is thermally and electrically an insulator or non-conductor. A magnetic field of

uniform strength is applied transversely to the direction of the flow and of the plate. The

magnetic field is assumed to be fixed with respect to the plate and the magnetic Reynolds

number of the flow is assumed to be small enough so that the induced magnetic field can be

neglected. Since no external electric field is applied and the effect of polarization of the ionized

fluid is negligible [22], it is assumed that the electric field is equal to zero.

Under the above assumptions, the equations governing this type of flow are the Reynolds-

Averaged Boundary Layer (RABL) equations which can be written for the MHD case, in the

absence of the body forces, in Cartesian coordinates ðx; yÞ, as follows:

Continuity equation:

@

@x�q�uþ q0u0� �

þ @

@y�q�vþ q0v0� �

¼ 0; ð1Þ

x-momentum equation:

�q�uþ q0u0� � @ �u

@xþ �q�vþ q0v0� � @ �u

@y¼ � @

�p

@xþ @

@yl@ �u

@y� �qu0v0 þ q0u0v0� �� �

� rB20 �u; ð2Þ

y-momentum equation:

@�p

@y¼ 0; ð3Þ

Energy equation:

cp ð�q�uþ q0u0Þ @�T

@xþ �q�vþ q0v0� � @ �T

@y

� �¼ @

@yk@ �T

@y� cp �qT0v0 � cpq0T0v0

� �

þ �u@ �p

@xþ l

@ �u

@y

� �2

þrB20 �uþ u0ð Þ2: ð4Þ

In the above equations we have replaced the instantaneous quantities f (e.g. u; v;T; q;p) by the

sum of their mean value (f ) and fluctuating parts (f 0), that is f ¼ f þ f 0. The last term in Eq. (2)

is the Lorentz force, whereas the last term in Eq. (4) is the Joule-heating term. These terms are

presented in the x-momentum and energy equations when a magnetic field is applied in the flow

u∞o

yMagnetic field B0

υw (suction/injection velocity)L

x

Edge (e)

Fig. 1. Flow configuration and coor-

dinate system

MHD compressible turbulent boundary-layer flow with adverse pressure gradient 173

field. In the absence of a magnetic field the above equations are reduced to the usual turbulent

boundary-layer flow equations [6].

Terms containing q0 can be dropped from the mass, momentum and energy equations for thin

shear layers. Also, the term q0u0 is negligible compared to q u as long as ðc� 1ÞM2 is not an order

ofmagnitude greater than unitywhereas the term q0v0 cannot be neglected, compared to q v, in the

continuity, momentum and energy equations [10]. The term rB20

�u02 in Eq. (4) is also neglected.On

the other hand, they-momentum equation (3) shows that the pressure variation is governed by the

free-stream and depends only on the coordinate x [14]. Using Bernoulli’s equation for the case of

magnetohydrodynamic flow [23], the term in the x-momentum equation can be substituted by

�dp

dx¼ qeue

due

dxþ rB2

0ue; ð5Þ

where the subscript e refers to the conditions at the edge of the boundary layer. Using the

abbreviation q v for q v þ q0v0 and omitting, for simplicity, the overbars on the basic time-

average variables u; v; q;p and T, the equations of the problem can now be written as

@

@xðquÞ þ @

@yðqvÞ ¼ 0; ð6Þ

qu@u

@xþ qv

@u

@y¼ qeue

due

dxþ @

@yl@u

@y� qu0v0

� �� rB2

0 u� ueð Þ; ð7Þ

qu@H

@xþ qv

@H

@y¼ @

@yk@T

@y� cpqT0v0 þ u l

@u

@y� qu0v0

� �� �: ð8Þ

It is worth mentioning here that the total enthalpy H for a perfect gas is defined by the

expression

H ¼ cpT þ 1

2u2: ð9Þ

Due to the parabolic nature of the above equations, boundary conditions must be provided

on two sides of the solution domain in addition to the initial conditions at x ¼ x0. So, the

boundary conditions of the problem under consideration are

y ¼ 0 : u ¼ 0; v ¼ vwðxÞ;@H

@y¼ 0;

y ¼ d : u ¼ ueðxÞ; H ¼ HeðxÞ;ð10Þ

where d is a distance sufficiently far away from the wall where the velocity u and total enthalpy

H reach their free-stream values, and vwðxÞ is the mass transfer velocity at the wall. In the case

of an impermeable wall vwðxÞ is equal to zero, for the case of suction vwðxÞ < 0 and for the case

of injection vwðxÞ > 0.

Defining the eddy kinematic viscosity em and turbulent Prandtl number Prt by the expressions

�u0v0 ¼ em

@u

@y; �T0v0 ¼ em

Prt

@T

@y; ð11Þ

the equations describing the problem can be written as

@

@xðquÞ þ @

@yðqvÞ ¼ 0; ð12Þ

qu@u

@xþ qv

@u

@y¼ qeue

due

dxþ @

@ylþ qemð Þ @u

@y

� �� rB2

0 u� ueð Þ; ð13Þ

174 M. Xenos et al.

qu@H

@xþ qv

@H

@y¼ @

@y

lPrþ q

em

Prt

� �@H

@yþ l 1� 1

Pr

� �þ qem 1� 1

Prt

� �� �u@u

@y

� �; ð14Þ

and the boundary conditions are

y ¼ 0 : u ¼ 0; v ¼ vwðxÞ;@H

@y¼ 0;

y ¼ d : u ¼ ueðxÞ; H ¼ HeðxÞ;ð15Þ

The above system of equations (12)–(15) consists of a coupled and nonlinear system of partial

differential equations (PDE) defined in the rectangular domain D ¼ fðx; yÞ : 0 < x < L;

0 < y <1g. In order to solve the system of PDEs numerically, the compressible version of the

Falkner-Skan transformation is introduced, defined by [10]

gðx; yÞ ¼Z y

0

ueðxÞmeðxÞx

� �1=2qðx; yÞqeðxÞ

dy; wðx; yÞ ¼ ðqeleuexÞ1=2f ðx; gÞ; ð16Þ

where f ðx; yÞ is the dimensionless stream function. Using the definition of the stream function wfor a compressible flow that satisfies the continuity equation (12), together with the relations

qu ¼ @w@y

; qv ¼ � @w@x

ð17Þ

and defining the dimensionless total energy ratio S as H=He; the system of the PDEs (12)–(15)

becomes

ðbf 00Þ0 þm1 ff 00 þm2½c� ð f 0Þ2� ¼ x m3 f 0 � 1ð Þ þ f 0@f 0

@x� f 00

@f

@x

� �; ð18Þ

ðeS0 þ df 0f 00Þ0 þm1 fS0 ¼ x f 0@S

@x� S0

@f

@x

� �; ð19Þ

g ¼ 0 : f 0 ¼ 0; fwðxÞ ¼ �1

ðueleqexÞ1=2

Zx

0

qwðx; 0ÞvwðxÞdx; S0w ¼ 0;

g ¼ ge : f 0 ¼ 1; S ¼ 1;

ð20Þ

where ge is the dimensionless thickness of the boundary layer. Primes denote partial differen-

tiation with respect to g. The quantities b, d, e, m1, m2, m3 etc. are defined as follows:

c ¼ qeðxÞqðx; gÞ ; C ¼ qðx; gÞlðx; gÞ

qeðxÞleðxÞ; b ¼ C 1þ eþm

� �;

d ¼ Cu2eðxÞ

HeðxÞ1� 1

Pr

� �þ eþm 1� 1

Prt

� �� �; e ¼ C

Pr1þ eþm

Pr

Prt

� �;

eþm ¼em

mðx; gÞ ; Rx ¼ueðxÞxmeðxÞ

;

m1 ¼1

21þm2 þ

x

qeðxÞleðxÞd

dxðqeleÞ

� �; m2 ¼

x

ueðxÞdueðxÞ

dx;

m3 ¼m0c

qeue

; m0 ¼ rB20:

ð21Þ

Finally, the problem under consideration is described by the system of equations (18) and (19),

subjected to the boundary conditions (20), where the coefficients entering into the equations are

defined by the expressions (21).

MHD compressible turbulent boundary-layer flow with adverse pressure gradient 175

3 Turbulent models

In aerodynamics as well as in other fields of fluid mechanics the need of calculating

compressible turbulent separated flows often appears. For this purpose several turbulent

models have been developed that can be used to represent eddy-kinematic viscosity eþm and

turbulent-Prandtl number Prt. In this study two algebraic turbulent models for the

calculation of the eddy-viscosity and a model for the turbulent-Prandtl number are

employed.

3.1 Cebeci-Smith turbulent model (C–S)

This model was developed by Cebeci and Smith [24] and is described in detail in [10]. It is

one of the most simple turbulent models and its accuracy has been explored and validated

for a wide range of experimental data. This model is one of the ‘‘Zero equation PDE

models’’, using only PDEs for the mean velocity field, and no turbulence PDEs [25]. For the

determination of the eddy-kinematic viscosity em only algebraic equations are involved. It

has been used for a wide range of engineering problems giving sufficiently accurate results

[6], [26].

The Cebeci-Smith turbulent model is a two-layer algebraic eddy viscosity model in which the

eddy-kinematic viscosity is given by

em ¼emð Þi; emð ÞiO emð Þo;

emð Þo; emð ÞiP emð Þo:

8<

:ð22Þ

According to the above formulation, the turbulent boundary layer is treated as a composite

layer consisting of inner and outer regions with separate expressions for the eddy-kinematic

viscosity in each region. For the inner region (viscous sublayer) the Prandtl-Van Driest for-

mulation is used while the Clauser formulation is involved for the outer region [10].

3.2 Baldwin-Lomax turbulent model (B–L)

Baldwin and Lomax improved the above turbulent model avoiding the necessity of finding the

edge of the boundary layer. It is an algebraic turbulent model that also treats the turbulent

boundary layer as a composite layer consisting of inner and outer regions. For the inner region

the Prandtl-Van Driest formulation is used. For the outer region, Baldwin and Lomax intro-

duced a new formulation according to which the product yMAXFMAX replaces d�ue in the

Clauser formulation of the Cebeci-Smith model and the combination yMAXU2DIF=FMAX replaces

dUDIF in the wake formulation [27].

The Baldwin-Lomax turbulent model was developed for use in two- or three-dimensional

Navier-Stokes machine codes. Such a code was developed by Steger [28], [29] and the results

from it are in a good agreement with the experimental data. Many researchers have opted the

Baldwin-Lomax algebraic model for its simplicity, although many modifications to its basic

form have been employed [30].

In the Baldwin-Lomax model we have adopted, from the Cebeci-Smith model, a formula for

the suction/injection velocity in order to investigate the mass transfer through the plate. So, we

did not consider the ‘‘damping-length’’ parameter Aþ as a constant taking the value 26, but as a

function of the local density and viscosity values [10].

176 M. Xenos et al.

3.3 Turbulent-Prandtl number

For the turbulent-Prandtl number Prt several expressions have been proposed. To the above

models the turbulent-Prandtl number is considered to be constant and equal to 0:9. In this

study a modification of the extended Kays and Crawford model is used [31]. More precisely Prt,

which can be used for all molecular Prandtl numbers, is given by the expression

Prt¼ 1

1

2 Prt1þ CPet

ffiffiffiffiffiffiffiffiffiffi1

Prt1

s

� CPetð Þ2 1� exp � 1

CPet

ffiffiffiffiffiffiffiffiffiffiPrt1p

� �� �( )

; ð23Þ

where Pet is the turbulent Peclet number and is given by the relation

Pet ¼ Pr eþm; ð24Þ

where C is a constant prescribing the spatial distribution of Prt versus Pet (C ¼ 0:3) and Prt1 is

the value of turbulent-Prandtl number far away from the wall [6].

4 Numerical solution

In order to investigate numerically the effect of an applied magnetic field on the flow field, a

numerical scheme must be applied. The numerical scheme used to solve the parabolic system of

PDEs (18)–(21) is a version of the Keller’s box method described in detail in [10] and [24] and

also in [6] and [32]. The scheme is unconditionally stable, and second-order accuracy is achieved

with nonuniform x and g spacing [33]. The governing equations are written as a first-order

system and derivatives of the unknown functions f ðx; gÞ, Sðx; gÞ with respect to g are intro-

duced as new functions. Using central difference derivatives for the unknown functions at the

midpoints of the net rectangle, the resulting difference equations are implicit and nonlinear. The

box-differencing scheme with Newton linearization is then applied to the first-order PDEs,

giving rise to a block tridiagonal system, which is solved by the block elimination method [34].

In most practical boundary-layer calculation involving a pressure gradient it is necessary to

predict the boundary layer over its whole length. That is, for a given external velocity distri-

bution and wall-temperature or heat-flux distribution and for a given transition point, it is

necessary to calculate the laminar, transitional and turbulent boundary layers, starting the

calculations at the leading edge (x ¼ 0). Starting from the leading edge, there is first a region

(0 < RX < Rxtr) in which the flow is laminar. After a certain distance there is a region

(Rxtr< RX < Rxt

) in which the flow is transitional, and in the third and last region (RX � Rxt)

the flow is fully turbulent.

In the work under consideration the calculations were started as laminar at x ¼ 0 and

transition was specified at xtr ¼ 0:066 m assuming that after this point the flow is fully tur-

bulent [6].

Table 1 presents some representative results concerning the separation point, the total drag

and the maximum temperature inside the boundary layer, for the Cebeci-Smith turbulent model

and for specific values of some parameters entering into the problem under consideration. It is

concluded that the obtained results are grid independent and a grid of 61 · 801 points is

sufficient to provide accurate numerical results. However, in our calculations a grid of

169 · 1601 points is used. The first number (169) indicates the number of points in g-directionwhereas the second (1601) is the number of points in x-direction.

MHD compressible turbulent boundary-layer flow with adverse pressure gradient 177

These representative numerical results are presented for two different Mach numbers

(M1 ¼ 1:5 and 3.0), for the case of no suction/injection and without application of a magnetic

field. The flow is considered to be adiabatic, the free stream temperature is T1 ¼ 300 K and the

Cebeci-Smith turbulence model is used. It is obvious that the obtained results concerning the

separation point, the total drag and the maximum temperature are almost identical and hence

grid-independent.

For the various parameters entering into the problem under consideration the following

considerations are made:

(i) The free-stream values for the viscosity l1, velocity u1, density q1 and total energy H1are calculated from the formulas below [10]:

l1 ¼ 1:45� 10�6T3=21 =ðT1 þ 110:33Þ; ð25Þ

u1 ¼ 20:04M1ffiffiffiffiffiffiffiT1

p; ð26Þ

q1 ¼ p1=287T1; ð27Þ

H1 ¼ cpT1 þ1

2u21 ð28Þ

for different values of the free-stream Mach number M1 and temperature T1, whereas the

edge values Te and pe were calculated by the formulas

Te ¼ T1 1� c� 1

2M21

ue

u1

� �2

�1

" #( )

; pe ¼ p1Te

T1

� �c=ðc�1Þ; ð29Þ

where c ¼ 1:4 and the edge values for viscosity, density and total energy were calculated

using formulas identical to those given by Eqs. (25), (27) and (28), respectively, except that

free stream values (1) were replaced by their edge values (e). Equation (25) is an inter-

polation formula for the calculation of air’s viscosity with temperature known as Suth-

erland’s law. In our study we adopted the previous interpolating formula since the

temperature differences are not small compared to the reference temperature due to high

velocity values and high temperatures that are applied over the plate.

(ii) The dimensionless heat-transfer parameter S0w is considered to be equal to zero, S0w ¼ 0

(S0ð0Þ ¼ 0), i.e., there is no heat transfer between the plate and the fluid (adiabatic flow).

In order to examine the influence of the applied magnetic field on the flow field, in the

presence of mass transfer, we study only the case of an adiabatic flow (S0w ¼ 0).

(iii) For determining the specific heat under constant pressure cp, the Prandtl numberPr and the

density q of the fluid (air) for temperatures varying from 100 to 2500 K, an interpolation

Table 1. Representative numerical results for separation point, the total drag and the maximum

temperature for different grids

Grid 1.5 Mach 3.0 Mach

Sep.

point (m)

Total

drag

Max.

temp (K)

Time

(s)

Sep.

point (m)

Total

drag

Max.

temp (K)

Time

(s)

61 · 801 4.634 790.5 411.0 4.5 5.123 6131.5 769.7 5.7

61 · 1601 4.637 792.7 411.0 16.8 5.122 6133.5 769.9 21.6

169 · 801 4.630 787.6 411.2 11.5 5.114 6111.7 770.1 14.3

169 · 1601 4.637 790.0 411.4 44.5 5.117 6114.1 770.4 54.5

178 M. Xenos et al.

formula is used. The data for cp,Pr andqwere taken from tables [10], [35]. The value of each

quantity for every temperature value is calculated by the successive linear interpolation

approach to high degree Lagrangian interpolation known as Neville’s algorithm. This

algorithm was selected because it is numerically stable, theoretically equivalent to the

Aitken’s algorithm but more efficient from a computational point of view [36]. In this study

we used a modification of the above algorithm, which takes into account only the 5 nearest

nodes of the temperature in question in order to accelerate the numerical calculation

without a significant loss of accuracy.

(iv) It can be assumed that the suction/injection velocity is constant over the plate and can be

applied to the whole length of the plate. The suction/injection velocity at the wall was

taken equal to vw ¼ �3:0� 10�4u1, which is a valid assumption to ensure that the flow

with suction/injection satisfies the simplifying conditions that form the basis of the

boundary-layer theory [14].

On the other hand, it is most interesting to apply a localized suction/ injection velocity to

a small slot over the plate near the point of separation for preventing the separation and

saving power that is required for the pumping. In order to examine the influence of the

localized suction/injection it is possible to apply a Gaussian distribution, given by the

expression [37]

vwðxÞ ¼ �Ase�ðx�xsÞ2=s2

; ð30Þ

where As is the suction/ injection strength, xs is the center of the slot and s is the half length of

the slot.

(v) To show the effect of the adverse pressure gradient on the flow field, as in [6] and [32] we

consider the linearly retarded flow, known as Howarth’s flow [38]. In the Howarth’s flow

the external velocity ue varies linearly with x, that is

ueðxÞ ¼ u1

�1� x

L

�: ð31Þ

In such a case, the dimensionless pressure gradient m2 is given by the relation m2 ¼ xx�L

. For

the numerical calculations the length L was taken equal to 8 m so that x varies from x ¼ 0 to 8

m.

(vi) In MHD boundary-layer problems the parameter m3 in Eqs. (21) is called ‘‘magnetic

parameter’’. This parameter is important since it represents the influence of the applied

magnetic field on the flow field. The parameter m0 is the product of the electrical con-

ductivity r and the square of the intensity B0 of the magnetic field which acts normal to

the plate and the flow (m0 ¼ rB20). In this study the intensity B0 of the magnetic field is

considered constant and its maximum value is equal to 10 or 12:5 Wb=m2 (Tesla). The

magnetic field is applied either to the whole length of the plate (global magnetic field) or to

a smaller part of the plate (local magnetic field). In order to avoid difficulties associated

with discontinuities to the interval where the magnetic field rises (from zero to the max-

imum intensity), simple smooth functions were introduced as

BðxÞ ¼ 1

2B0 1þ tanh b x� að Þ½ �; 0 < x � aþ b

2ð32Þ

and

BðxÞ ¼ 1

2B0 1� tanh b x� bð Þ½ �; x >

aþ b

2; ð33Þ

MHD compressible turbulent boundary-layer flow with adverse pressure gradient 179

where a; b is the beginning and the end of the applied magnetic field, b ¼ 10 and B0 is the

maximum intensity applied.

The electrical conductivity is calculated from the relation [39]

r ¼hrT

lB

lb0

� h

hB

�niþ a; ð34Þ

where rT is the reference electrical conductivity and is taken to be 100 mho/m. The

constant b0 determines the pressure dependence of r. So, for pressure p ¼ 1 atm the

constant is equal to b0 � 5:68� 10�8. For rarefied gases, e.g., air in very high altitude

where the pressure is much smaller (p ¼ 10�3), this constant is an order of magnitude

smaller, b0 � 1:20� 10�9. The h is the specific enthalpy (h ¼ cpT), l the viscosity of air

and the subscript B refers to the temperature of 222 K. The superscript n is equal to 4:79

and a is a constant that describes the increment of r due to seeding of an ion in the flow

field [9], [13], [16] or due to other non-equilibrium ionization methods [17]. It is worth

mentioning here that for high free-stream temperatures the constant a can be considered

equal to zero.

For the numerical solution of equations describing the problem under consideration we

refer that the program is divided in two parts. The first and most vital part is a Dynamic

Link Library (DLL) which contains all the algorithms for the numerical evaluation of the

problem. So this part is the core of the whole program and contains the construction of the

grid above the plate, the descretized PDEs with the Keller box method, the block tridiagonal

elimination method for the algebraic solution of the discretized PDEs, etc. The second part is

essentially a graphical user interface (GUI), where the user can review or alter the initial data,

as the free stream Mach number, the temperature of the plate and the fluid, the suction/

injection velocity, the magnetic parameter, etc. The user can also choose a turbulent model

(Cebeci-Smith or Baldwin-Lomax) from the GUI and can alter the x and g step. Finally, he

can review and manipulate the visualization of the data produced from the numerical inte-

gration in the DLL. The program was written in FORTRAN 90 utilizing OpenGL� for the

visualization of the data [40].

5 Results and discussion

It is very interesting to examine the dimensional quantities of the problem under consideration,

as the dimensional velocity and the dimensional temperature, at each grid point ðx; yÞ on the

flow field of the compressible boundary layer. The investigation of the above dimensional

quantities gives a clear image of the shape of the boundary layer under the adverse pressure

gradient which is very different from the one without pressure gradient. It is also very inter-

esting to examine the skin friction coefficient and the total drag on the plate, at each point ðx; 0Þon the plate. The two turbulent models (C-S, B-L turbulent models) give qualitatively same

results in all the quantities examined. But each model gives a different separation point and

quantitatively the results differ from model to model.

In aerodynamics the most important quantities for engineering applications are the skin

friction coefficient Cf xand the local Standon number Stx for the cases of a heating or cooling

wall (S0w 6¼ 0 or Sw 6¼ 1). So, in order to show the effect of an applied magnetic field (local or

global) or the combined effect of the applied magnetic field and the localized suction/injection

180 M. Xenos et al.

velocity on the compressible boundary-layer flow, we must investigate these two quantities. The

above quantities are the most frequently examined and can be written as [6]

Cfx ¼2Cwffiffiffiffiffiffi

Rx

p f 00w; Stx ¼CwS0w

PrffiffiffiffiffiffiRx

p1� Swð Þ

Sw 6¼ 1ð Þ; ð35Þ

where f 00w ¼ f 00ðx; 0Þ is the dimensionless wall-shear parameter, S0w ¼ S0ðx; 0Þ is the dimension-

less wall heat-transfer parameter and Sw ¼ Hw=He is the dimensionless total-enthalpy ratio on

the wall. Another very important quantity is the total drag D on the plate. It can be defined, per

unit width of the plate, by the integral [9]

D ¼Z x�

0

Cwf 00wðx; 0ÞffiffiffiffiRp

x

qeðxÞu2eðxÞdx; ð36Þ

where Cw ¼ qwlw=qele is a function of x and x� is the separation point.

We have examined the influence of the magnetic field at low free-stream temperature (300 K)

with constant electrical conductivity (in this study r for the low-temperature condition is

considered equal to 1.0 mho/m) and at high free-stream temperature (1000 K) with conductivity

r as a function of temperature T (Eq. (34)). The free stream Mach number is considered

constant for all cases examined and equal to Mach 3.0 with the plate adiabatic (S0w ¼ 0). So,

there is no heat transfer between the plate and the fluid. This permits to examine the net

influence of the magnetic field on the flow field. However, due to high free-stream velocity the

viscous heating inside the boundary layer is significant.

5.1 Low free-stream temperature (300 K)

The case of low free-stream temperature, where the electrical conductivity is considered to be

constant and equal to 1.0 mho/m, is examined. The dimensional velocity field in the turbulent

boundary-layer for the cases of (a) B0 ¼ 0 (no magnetic field, no suction) and (b) B0 = 10 Tesla

(magnetic field and localized suction) are shown in Figs. 2 and 3 (for both cases M1 ¼ 3:0, r =

1.0 mho/m and the C-S turbulent model). The maximum suction velocity is equal to

vw ¼ �3:0� 10�4 u1, the middle of the slot position lies at x = 4.8 m and the half slot length

is 0.7 m. It is observed that the adverse pressure gradient considerably increases the thickness of

the boundary layer and this increment continues until the point of separation. In the region of

separation the velocity near the wall is almost zero and the direction of the vectors is orientated

to the edge of the boundary layer. This shows that the fluid particles cannot penetrate too far

into the region of increased pressure due to their small kinetic energy. The above is obvious for

both cases, but in the case of magnetic field and localized suction the separation point moves

downstream to the tip of the plate.

For more details, zooming near the wall, from x = 4.0 m to 5.0 m (near the region of

separation), we observe more clearly the influence of the magnetic field and localized suction.

In case (a) (no magnetic field, no suction), the direction of the velocity vectors is orientated to

the edge of the boundary layer, while in case (b) (magnetic field and localized suction), the

vectors’ direction is orientated to the wall in the inner sublayer, and is almost parallel to the

wall in the upper levels. Hence, it is concluded that the combined effect of the magnetic field

and the localized suction maintains the turbulent boundary layer on the plate, delaying

separation.

The application of a magnetic field and suction has minimal effects on the boundary-layer

temperature (Fig. 4, C-S turbulent model). So, the maximum temperature in case (a) (no

magnetic field and suction) is 770 K and in case (b) is 781 K, the difference in the maximum

MHD compressible turbulent boundary-layer flow with adverse pressure gradient 181

temperature being insignificant. It is worth mentioning here, however, that the fluid tempera-

ture inside the boundary layer increases significantly along the plate. All the above results

concern the C-S turbulent model. The respective results for the B-L model are qualitatively

alike but due to lack of space we do not indicate them in the results.

Case (a)0.12D+00

0.10D+00

0.75D–01

0.50D–01

0.25D–01

xy

0.10D+010.20D+010.30D+010.40D+010.50D+010.60D+01 0.43D+01 0.44D+01 0.46D+01 0.48D+010.51D+01

0.13D–01

0.12D–01

0.10D–01

0.86D–02

0.70D–02

0.54D–02

0.38D–02

0.22D–02

0.67D–03

Fig. 2. Vector field of the dimensional velocity for the case (a): no magnetic field and no suction (300 K)

0.10D+01 0.20D+01 0.30D+01 0.40D+01 0.50D+01 0.60D+010.53D+01

0.47D+01 0.48D+01 0.50D+01 0.51D+01

Case (b)

xy

0.15D+00

0.12D+00

0.10D+00

0.75D–01

0.50D–01

0.25D–01

0.11D–01

0.97D–02

0.83D–02

0.69D–02

0.55D–02

0.41D–02

0.28D–02

0.14D–02

–0.96D–05

Fig. 3. Vector field of the dimensional velocity for the case (b): global magnetic field and localizedsuction (300 K)

0.10D+01 0.20D+01 0.30D+01 0.40D+01 0.50D+01 0.60D+010.51D+01

0.10D+01 0.20D+01 0.30D+01 0.40D+010.50D+010.60D+010.53D+01

Case (a)

xy

Case (b)

xy

0.78+03

273.0

0.00D+00

0.15D+00

0.12D+00

0.10D+00

0.75D–01

0.50D–01

0.25D–01

0.78+03

273.0

0.00D+00

0.12D+00

0.10D+00

0.75D–01

0.50D–01

0.25D–01

Fig. 4. Temperature for the cases of (a) no magnetic field, no suction and (b) magnetic field andlocalized suction

182 M. Xenos et al.

The skin friction coefficient Cf xversus the distance x for the two turbulence models and for

the cases of no magnetic field, no suction (1), magnetic field, no suction (2) and the combination

of magnetic field and localized suction (3) is shown in Fig. 5. It is reminded that the action of

the magnetic field is to the whole length of the plate (global magnetic field). From this figure the

effect of the magnetic field and localized suction on the flow field, on the local skin friction

coefficient Cf xand the separation point is, once more, confirmed. The application of a magnetic

field shifts the separation point downstream the plate, from x = 5.12 m (no magnetic field, no

suction) to x = 5.25 m (curves (1), (2)). The additional effect of localized suction shifts the

separation point further downstream to x = 5.35 m for the C-S model (curve (3)). For the B-L

model the separation point with no magnetic field is at x = 5.76 m, with magnetic field at x =

5.85 m and with the combined influence of the localized suction the separation point moves

further downstream at x = 5.94 m. The values of total drag D for each case are also shown in

this figure. We can conclude that the presence of the magnetic field and the localized suction

increases frictional drag. However, we expect that this increment is small compared to the

increase in friction drag if separation had occurred, since after separation large energy losses

and drag increment take place due to flow reversal.

0.10D+01 0.20D+01 0.30D+01 0.40D+01 0.50D+01Distance x(m)

0.10D+01 0.20D+01 0.30D+01 0.40D+01 0.50D+01Distance x(m)

0.25D+01

0.20D+01

0.15D+01

0.10D+01

0.50D+01

Skin

fri

ctio

n co

effi

cien

t

0.25D+01

0.20D+01

0.15D+01

0.10D+01

0.50D+01

Skin

fri

ctio

n co

effi

cien

t

(1) D= 6114.13(2) D= 6407.55(3) D= 6768.00

(1) D= 7472.76(2) D= 7696.10(3) D= 8273.24

Minf = 3.0Baldwin-Lomax turbulent modelT= 300 K

(1) No magnetic field, no localized suction(2) Magnetic field, no localized suction(3) Magnetic field, localized suction

Minf = 3.0Cebeci-Smith turbulent modelT= 300 K

(1) No magnetic field, no localized suction(2) Magnetic field, no localized suction(3) Magnetic field, localized suction

(1)(2)

(3)(1)

(2)

(3)

Fig. 5. Variations of the skin friction coefficient Cfx for the two turbulent models (C-S and B-L) and forglobal magnetic field (300 K)

0.10D+01 0.20D+01 0.30D+01 0.40D+01 0.50D+01Distance x(m)

0.10D+01 0.20D+01 0.30D+01 0.40D+01 0.50D+01Distance x(m)

0.25D+01

0.20D+01

0.15D+01

0.10D+01

0.50D+01

Skin

fri

ctio

n co

effi

cien

t

0.25D+01

0.20D+01

0.15D+01

0.10D+01

0.50D+01

Skin

fri

ctio

n co

effi

cien

t

Minf = 3.0Cebeci-Smith turbulent modelT= 300 K

(1) No magnetic field, no localized suction(2) Local magnetic field, no localized suction(3) Local magnetic field, localized suction

Minf = 3.0Baldwin-Lomax turbulent modelT= 300 K

(1) No magnetic field, no localized suction(2) Local magnetic field, no localized suction(3) Local magnetic field, localized suction

(1) D= 7472.76(2) D= 7566.27(3) D= 8135.22

(1) D= 6114.13(2) D= 6203.00(3) D= 6514.20

(1)(2)

(3)(1)

(2)

(3)

Fig. 6. Variations of the skin friction coefficient Cfx for the two turbulent models (C-S and B-L) and for

local magnetic field (300 K)

MHD compressible turbulent boundary-layer flow with adverse pressure gradient 183

It is also very interesting to examine the influence of a local magnetic field on the skin

friction coefficient. The application of a local magnetic field is more desirable, because it is

easier and more economic to apply a magnetic field to a specific length instead of to the

whole length of the plate. The magnetic field is applied to a length of 3 m, from x = 3.0 m to

x = 6.0 m; this region is the area near the separation point. The boundary-layer in this area,

due to the adverse pressure gradient, is very unstable. So, the application of the magnetic

0.25D+01

0.20D+01

0.15D+01

0.10D+01

0.50D+01

Skin

fri

ctio

n co

effi

cien

t

0.10D+01 0.20D+01 0.30D+01 0.40D+01 0.50D+01 0.60D+01Distance x(m)

(1) D= 6550.70(2) D= 6407.55(3) D= 6262.44(4) D= 6114.13

Minf = 3.0

Cebeci-Smith turbulent model(no suction)

T= 300 K

(1) Magnetic field B0 = 12.6 Tesla(2) Magnetic field B0 = 10 Tesla(3) Magnetic field B0 = 7 Tesla(4) No magnetic field

(1)

(2)(3)

(4) Fig. 8. Variations of the skin frictioncoefficient Cfx for the Cebeci–Smith

turbulent model with no suction andmagnetic field (300 K)

0.25D+01

0.20D+01

0.15D+01

0.10D+01

0.50D+01

Skin

fri

ctio

n co

effi

cien

t

0.10D+01 0.20D+01 0.30D+01 0.40D+01 0.50D+01 0.60D+01Distance x(m)

(1) D= 6408.60(2) D= 6297.40(3) D= 6201.75(4) D= 6114.13

Minf = 3.0

Cebeci-Smith turbulent model(no magnetic field)

T= 300 K

(1) Localized suction (υw = –3.0 10–4 unif)(2) Localized suction (υw = –2.0 10–4 unif)(3) Localized suction (υw = –1.0 10–4 unif)(4) No suction

(1)

(2)(3)

(4)

Fig. 7. Variations of the skin friction

coefficient Cfx for the Cebeci–Smithturbulent model with no magnetic

field and localized suction (300 K)

0.25D+01

0.30D+01

0.35D+01

0.40D+01

0.20D+01

0.15D+01

0.10D+01

0.50D+01

Skin

fri

ctio

n co

effi

cien

t

0.10D+01 0.20D+01 0.30D+01 0.40D+01 0.50D+01 0.60D+01Distance x(m)

(1) D= 790.05(2) D= 870.87(3) D= 6114.13(4) D= 6262.44

Minf = 1.5

Cebeci-Smith turbulent model

(1) No magnetic field(2) Magnetic field B0 = 7 Tesla

Minf = 3.0(3) No magnetic field(4) Magnetic field B0 = 7 Tesla

T= 300 K(no suction)

(1)

(2)(3)

(4)

Fig. 9. Variations of the skin frictioncoefficient Cfx for the Cebeci–Smith

turbulent model with no suction andwith (or without) magnetic field

(300 K)

184 M. Xenos et al.

field in this region stabilizes the boundary layer. Minor influences of the local magnetic field

in a region away from the separation point were observed (e.g., from x = 0.0 m to x =

3.0 m). In Fig. 6, the effect of the local magnetic field and the localized suction on the skin

friction coefficient Cf xand the separation point is presented. More precisely, the local

magnetic field shifts the separation point downstream by 9 cm to x = 5.21 m and the

additional influence of the localized suction moves the separation point 9 cm more, to x =

5.30 m for the C-S model. Using the B-L model, the separation point moves downstream

with the application of the local magnetic field for 8 cm, to x = 5.84 m, with the additional

influence of the localized suction moving the separation point another 8 cm downstream, to x

= 5.92 m. It is evident, that the local magnetic field influences the flow field almost as much

as the global magnetic field. The values of total drag D for each case are also shown in this

figure. The values of the total drag are smaller than those in the previous case (global

magnetic field).

The dimensionless skin friction coefficient Cfx is the most important quantity for engi-

neering applications and, on the other hand, suction/injection as well as magnetic field can

affect this quantity. So, some additional calculations are carried out for this quantity and

the obtained results are presented in Figs. 7–9. It is observed that increment of the strength

of localized suction velocity vw counteracts the adverse pressure gradient (Fig. 7) but in-

creases the total drag as the flow remains on the plate for higher values of jvwj. Also, an

increase of the magnetic parameter shifts the separation point downstream the plate. A

magnetic field of a strength of 10 Tesla is necessary to affect the velocity field whereas less

strong fields (< 5 Tesla) have negligible or very small effects for such a high Mach number.

It is also observed an increment of the total drag when the magnetic parameter increases

(Fig. 8). This is also due to the fact that the flow remains on the plate for larger distances as

B0 increases.

For higher Mach numbers, larger magnetic parameter (a stronger magnetic field) should be

applied in order to shift the separation point. Application of the same magnetic parameter

(B0 ¼ 7) on less convective flows (Mach 1.5) shifts the separation point farther downstream

compared to highly convective flows (Mach 3) (Fig. 9).

5.2 High free-stream temperature (1000 K)

For the case of high free-stream temperature, where the conductivity r is considered as a

function of temperature T, we give only results for the velocity field, the skin friction coef-

ficient and total drag. The obtained results for the velocity field are shown only for the B-L

turbulent model. The C-S model gives qualitatively the same results, not indicated here due to

lack of space. The dimensional velocity field above the flat plate for the cases of (a) B0 ¼ 0

(no magnetic field, no suction), and (b) B0 = 10 Tesla and localized suction are shown in

Figs. 10 and 11 (for both cases (a) and (b) and for M1 ¼ 3:0, r = rðTÞ and B-L turbulent

model). The maximum suction velocity is equal to vw ¼ �3:0� 10�4 u1, the middle of the

slot position lies at x = 5.1 m and the half slot length is 0.7 m. It is observed that the

adverse pressure gradient retards the fluid particles over the plate and the turbulent boundary

layer finally separates from the plate. The separation point is shifted downstream in this case

compared to the previously described (300 K, seeding). The magnetic field delays the sepa-

ration, stabilizing the boundary-layer, but it can’t override the influence of the adverse

pressure gradient. The additional effect of the localized suction shifts the separation point

more downstream to the tip of the plate.

MHD compressible turbulent boundary-layer flow with adverse pressure gradient 185

Zooming near the wall, from x = 4.5 m to 5.5 m (near the region of separation), we observe

more clearly the influence of the magnetic field and localized suction. The result is almost the

same as the one in the previous case (300 K).

The skin friction coefficient Cf xagainst the distance x for the two turbulent models and for

the cases of (1) no magnetic field, no suction, (2) magnetic field, no suction, and (3) the

combination of magnetic field and localized suction are shown in Fig. 12. From this figure,

the effect of the magnetic field and localized suction on the local skin friction coefficient Cf x

and the separation point is depicted. The application of a magnetic field (without suction)

shifts the separation point downstream the plate, from x = 5.34 m (no magnetic field, no

suction) to x ¼ 5.58 m. The additional effect of localized suction moves the separation point

further downstream to x = 5.68 m for the C-S model. For the B-L model the separation

point with no magnetic field and no suction is at x = 5.83 m, with magnetic field (without

suction) the separation point moves downstream to x = 6.08 m and with the combined

influence of the localized suction the separation point moves further downstream to x =

6.17 m. The magnetic field influence on the boundary layer is more evident in this case

(1000 K) than the previous one (300 K). The values of total drag D for each case are also

shown in this figure. The presence of the magnetic field and the localized suction increases

frictional drag. However, we may once again expect that this increment is small compared to

Case (a)

0.12D+00

0.15D+00

0.17D+00

0.10D+00

0.75D–01

0.50D–01

0.25D–01

xy

0.10D+01 0.20D+01 0.30D+01 0.40D+010.50D+01 0.60D+01 0.47D+01 0.49D+01 0.51D+01 0.53D+01 0.550.58D+01

0.12D–01

0.11D–01

0.94D–02

0.80D–02

0.65D–02

0.51D–02

0.36D–02

0.22D–02

0.69D–03

Fig. 10. Vector field of the dimensional velocity for the case (a) no magnetic field and no suction(1000 K)

Case (b)

0.12D+00

0.15D+00

0.17D+00

0.20D+00

0.10D+00

0.75D–01

0.50D–01

0.25D–01

xy

0.10D+01 0.20D+01 0.30D+010.40D+01 0.50D+01 0.60D+01 0.47D+01 0.49D+01 0.51D+01 0.53D+010.62D+01

0.10D–01

0.11D–01

0.86D–02

0.72D–02

0.58D–02

0.44D–02

0.30D–02

0.16D–02

0.23D–03

Fig. 11. Vector field of the dimensional velocity for the case (b) global magnetic field and localizedsuction (1000 K)

186 M. Xenos et al.

the increase in friction drag if separation had occurred. The value of the total drag in this

case (1000 K) due to high free-stream temperature is greater than the respective value in the

case of low free-stream temperature (300 K).

The local magnetic field is applied near the separation point, to a length of 3 m, from x =

3.0 m to x = 6.0 m. The boundary layer in this area, due to the adverse pressure gradient, is

very unstable and the application of the magnetic field in this region stabilizes the boundary-

layer. In Fig. 13, the effect of the local magnetic field and the localized suction on the skin

friction coefficient Cf xand the separation point is presented. More precisely, the local magnetic

field shifts the separation point downstream by 23 cm to x = 5.57 m and the additional

influence of the localized suction moves the separation point 9 cm more to x = 5.66 m for the

C-S model. For the B-L model, the separation point moves downstream with the application of

the local magnetic field by 21 cm, to x = 6.04 m, and the additional influence of the localized

suction moves the separation point further downstream to x = 6.12 m (8 cm). It is evident that

the local magnetic field influences the flow field almost as much as the global magnetic field.

The values of total drag D for each case are also shown in this figure. The values for the total

drag are smaller than those in the previous case (global magnetic field).

0.10D+01 0.20D+01 0.30D+01 0.40D+01 0.50D+01

Distance x(m)

0.10D+01 0.20D+01 0.30D+01 0.40D+01 0.50D+01 0.60D+01Distance x(m)

0.25D+01

0.30D+01

0.35D+01

0.20D+01

0.15D+01

0.10D+01

0.50D+00

Skin

fri

ctio

n co

effi

cien

t

0.25D+01

0.30D+01

0.35D+01

0.20D+01

0.15D+01

0.10D+01

0.50D+00

Skin

fri

ctio

n co

effi

cien

tMinf = 3.0Cebeci-Smith turbulent modelT= 1000 K

(1) No magnetic field, no localized suction(2) Magnetic field, no localized suction(3) Magnetic field, localized suction

Minf = 3.0Baldwin-Lomax turbulent modelT= 1000 K

(1) No magnetic field, no localized suction(2) Magnetic field, no localized suction(3) Magnetic field, localized suction

(1) D= 8090.57(2) D= 8319.28(3) D= 8694.91

(1) D= 9342.63(2) D= 9588.11(3) D= 10149.65(1)

(2)(3)

(1)(2)

(3)

Fig. 12. Variations of the skin friction coefficient Cfx for the two turbulent models (C-S and B-L) and forglobal magnetic field (1000 K)

0.10D+01 0.20D+01 0.30D+01 0.40D+01 0.50D+01Distance x(m)

0.10D+01 0.20D+01 0.30D+01 0.40D+01 0.50D+01 0.60D+01Distance x(m)

0.25D+01

0.30D+01

0.35D+01

0.20D+01

0.15D+01

0.10D+01

0.50D+00

Skin

fri

ctio

n co

effi

cien

t

Cebeci-Smith turbulent model

0.25D+01

0.30D+01

0.35D+01

0.20D+01

0.15D+01

0.10D+01

0.50D+00

(3)

(3)

(2)

(1)(2)

(1)

Skin

fri

ctio

n co

effi

cien

t

Minf = 3.0Baldwin-Lomax turbulent modelT= 1000 K

(1) No magnetic field, no localized suction(2) Local magnetic field, no localized suction(3) Local magnetic field, localized suction

Minf = 3.0

T= 1000 K

(1) No magnetic field, no localized suction(2) Local magnetic field, no localized suction(3) Local magnetic field, localized suction

(1) D= 8090.57(2) D= 8275.13(3) D= 8631.00

(1) D= 9342.63(2) D= 9554.20(3) D= 10109.82

Fig. 13. Variations of the skin friction coefficient Cfx for the two turbulent models (C-S and B-L) and forlocal magnetic field (1000 K)

MHD compressible turbulent boundary-layer flow with adverse pressure gradient 187

In the case of high free-stream temperature (T = 1000 K), the same effects of the magnetic

field on the flow are observed as in the low free-stream temperature case. Increment of the

magnetic field counteracts the adverse pressure gradient, retains the flow field, shifts the sep-

aration point downstream the plate and increases the total drag. The above results are more

evident for higher magnetic fields (Fig. 14). Finally, in both cases of low and high free-stream

temperature it is observed that the localized suction has the same effect on the flow field as the

magnetic field. Localized suction retains the flow on the plate for longer distances, moves the

separation point farther downstream the plate and increases the total drag. These effects are

more pronounced as the suction parameter increases (Fig. 15).

6 Conclusions

– The magnetic field for both cases (low and high free-stream temperature) influences the

turbulent boundary-layer, shifts the separation point downstream to the tip of the plate and

increases the total drag, and this influence of the magnetic field on the flow field is greater in

the case of the high free-stream temperature.

– The application of a local magnetic field near the separation point influences the flow field

almost as much as the application of a global magnetic field.

0.25D+01

0.30D+01

0.35D+01

0.20D+01

0.15D+01

0.10D+01

0.50D+00

Skin

fri

ctio

n co

effi

cien

t

0.10D+01 0.20D+01 0.30D+01 0.40D+01 0.50D+01 0.60D+01Distance x(m)

(1) D= 8446.56(2) D= 8319.28(3) D= 8200.93(4) D= 8090.57

Minf = 3.0

Cebeci-Smith turbulent model(no suction)

T= 1000 K

(1) Magnetic field B0 = 12.6 Tesla(2) Magnetic field B0 = 10 Tesla(3) Magnetic field B0 = 7 Tesla(4) No magnetic field

(1)(2)

(3)(4)

Fig. 14. Variations of the skin friction

coefficient Cfx for the Cebeci–Smithturbulent model with no suction and

magnetic field (1000 K)

0.25D+01

0.30D+01

0.35D+01

0.20D+01

0.15D+01

0.10D+01

0.50D+00

Skin

fri

ctio

n co

effi

cien

t

0.10D+01 0.20D+01 0.30D+01 0.40D+01 0.50D+01 0.60D+01Distance x(m)

(1) D= 8359.04(2) D= 8263.83(3) D= 8173.58(4) D= 8090.60

Minf = 3.0

Cebeci-Smith turbulent model(no suction)

T= 1000 K

(1)(2)

(3)(4)

(1) Localized suction (υw = –3.0 10–4 unif)(2) Localized suction (υw = –2.0 10–4 unif)(3) Localized suction (υw = –1.0 10–4 unif)(4) No suction

Fig. 15. Variations of the skin friction

coefficient Cfx for the Cebeci–Smithturbulent model with no magnetic

field and localized suction (1000 K)

188 M. Xenos et al.

– The additional effect of localized suction shifts the separation point farther downstream to

the tip of the plate, increasing total drag. The influence of the localized suction is the same

for both cases.

– Both turbulent models (C-S and B-L) give qualitatively the same results. For both models

the influence of the magnetic field (local or global) and localized suction are the same,

moving the separation point downstream and stabilizing the boundary layer.

– The influence of the magnetic field and localized suction on the temperature is insignificant.

– An increment of the magnetic field shifts the separation point farther downstream to the tip

of the plate, increasing the total drag.

– An increment of the localized suction shifts the separation point farther downstream to the

tip of the plate, increasing the total drag.

– For low free stream temperature the influence of the magnetic field decreases as the Mach

number increases.

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Authors’ address: M. Xenos, S. Dimas and N. Kafoussias, Department of Mathematics, Division of

Applied Analysis, University of Patras, 26500 Patras, Greece (E-mail: nikaf@math.upatras.gr)

190 M. Xenos et al.: MHD compressible turbulent boundary-layer flow with adverse pressure gradient