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Unsteady compressible boundary layer flow over a circular cone near a plane of symmetry

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Abstract

An analysis has been performed to study the unsteady laminar compressible boundary layer governing the hypersonic flow over a circular cone at an angle of attack near a plane of symmetry with either inflow or outflow in the presence of suction. The flow is assumed to be steady at time t=0 and at t>0 it becomes unsteady due to the time-dependent free stream velocity which varies arbitrarily with time. The nonlinear coupled parabolic partial differential equations under boundary layer approximations have been solved by using an implicit finite-difference method. It is found that suction plays an important role in stabilising the fluid motion and in obtaining unique solution of the problem. The effect of the cross flow parameter is found to be more pronounced on the cross flow surface shear stress than on the streamwise surface shear stress and surface heat transfer. Beyond a certain value of the cross flow parameter overshoot in the cross flow velocity occurs and the magnitude of this overshoot increases with the cross flow parameter. The time variation of the streamwise surface shear stress is more significant than that of the cross flow surface shear stress and surface heat transfer. The suction and the total enthalpy at the wall exert strong influence on the streamwise and cross flow surface shear stresses and the surface heat transfer except that the effect of suction on the cross flow surface shear stress is small.

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Abbreviations

a :

Velocity of sound, ms−1

A :

Dimensionless suction parameter=−(3/2)1/2 [(ρ w)w/ρeu0] Re 1/2 x

C p :

Constant pressure specific heat, J kg−1 K

C v :

Constant volume specific heat, J kg−1 K

Ec:

Viscous dissipation parameter=u2 0/2He

f ′:

Dimensionless velocity component along streamwise direction=u/ue

g :

Dimensionless total enthalpy=H/He

h :

Specific enthalpy, J kg−1

H :

Total enthalpy, J kg−1

k :

Fluid thermal conductivity, W m−1 K

L :

Denotes dimensionless dependent variable f ′ or s ′ or g

Me:

Mach number at the edge of the boundary layer=V/a

N :

Product of the density–viscosity ratio=ρμ/ρeμe

p :

Static pressure, Pa

p 0 :

Static pressure when θ=0, Pa

p 2 :

Denotes the curvature of the pressure distribution along the plane of symmetry, Pa

Pr:

Prandtl number=μeC p /k

r :

Cylindrical radius of the cone, m

R :

Dimensionless function of dimensionless time=1+ɛτ2

Re x :

Reynolds number=u0x/ νe

s ′:

Dimensionless cross flow velocity profile=v/ve

t :

Time, s

T :

Temperature, K

u,v,w:

Velocity components along x,θ and z directions, respectively, ms−1

u0,v0:

Value of u and v at time t=0, ms−1

V :

Fluid velocity in the inviscid flow, ms−1

x :

Distance along a generator of the cone from apex, m

z :

Distance normal to the surface, m

α:

Dimensionless cross flow parameter=2ξ v0/(ρeμeu 20 r3)

α0:

Angle of attack

β:

Dimensionless parameter associated with the three-dimensional nature of the flow=(2ξ/v0 x) d(ν0 x)/dξ

γ:

Specific heat ratio=cp/c_v (1.4 for air)

Δη,Δτ:

Step sizes in η and τ directions, respectively

η:

Transformed coordinate normal to the surface=(3/2)(ρeu0/μex)1/2\( {\int_0^z {{\left( {\rho /\rho _{e} } \right)}} }dz \)

θ:

Circumferential angle measured from plane of symmetry

θc:

Semi-vertical angle of the cone

μ:

Viscosity coefficient, kg m−1 s−1

ν:

Kinematic viscosity, m2 s−1

ξ:

Transformed streamwise coordinate=3−1 ρeμeu0 (sin θc)2x3

ρ:

Mass density, kg m−3

τ:

Dimensionless time=(3/2)(u0/x)t

ω:

Index in the power-law variation of viscosity coefficient

e:

Conditions at the edge of the boundary layer

i:

Initial conditions

w:

Wall conditions

′:

Prime denotes derivative with respect to η

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Authors and Affiliations

  1. Manufacturing Engineering Department, The Public Authority for Applied Education& Training, Shuweikh, 70654, Kuwait

    A. J. Chamkha

  2. Department of Engineering, Manchester Metropolitan University, Manchester, M1 5GD, U.K

    H. S. Takhar

  3. Department of Mathematics, Indian Institute of Science, Bangalore, 560012, India

    G. Nath

Authors
  1. A. J. Chamkha
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  2. H. S. Takhar
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  3. G. Nath
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Correspondence to H. S. Takhar.

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Chamkha, A.J., Takhar, H.S. & Nath, G. Unsteady compressible boundary layer flow over a circular cone near a plane of symmetry. Heat Mass Transfer 41, 632–641 (2005). https://doi.org/10.1007/s00231-004-0575-8

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  • DOI: https://doi.org/10.1007/s00231-004-0575-8

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