Measurements in the vicinity of a stagnation point

This paper presents measurements of a plane jet impinging onto a normal flat plate placed up to five jet widths from the jet outlet. The small spacing ensured that the stagnation streamline remained in the potential core of the jet. The plate shear stress distribution compared well to that from an analytical solution for the laminar development of the plate boundary layer whose external velocity was determined from the measured pressure. By comparing the shear stress measured under the present low level of free stream turbulence (0.35%) at the jet exit with that of Tu and Wood [Exp. Thermal Fluid Sci. 13 (1996) 364–373] made at about 4%, it is concluded that the turbulence level at the nozzle exit has only a second-order influence on the surface shear stress around the stagnation point. Some spanwise non-uniformity was observed in the plate shear stress, but this was confined largely to the transition region. The mean velocity, Reynolds stresses, and fluctuating pressure were measured along the stagnation streamline using a fast-response pressure probe. A significant increase in the streamwise normal stress and the mean square of the pressure fluctuations occurred before they were eventually attenuated by the plate. This increase occurred in the region where the streamwise velocity was decreasing close to the plate causing extra energy production through the normal stresses. Spectra of the velocity and pressure fluctuations showed that the increase in level was mainly due to the low frequency motion, whereas the subsequent decrease occurred at higher frequencies.     

Effect of a magnetic field on the flow in the vicinity of the stagnation point of a blunt body with ablation of a protective layer

During hypersonic flow around a blunt-nosed body, the gas which passes through the bow shock is heated to high temperatures, where dissociation, ionization, and inverse phenomena (recombination) take place in the gas. If an ionized gas moves in a magnetic field, the ponderomotive force which is set up changes the nature of its motion close to the stagnation point, decreasing the frictional stress and heat transfer at the wall (at the contact surface of the gas and the body about which the gas flows). In this case, the intense heat fluxes from the strongly heated gas to the body about which the gas flows cause phase changes in the surface of the body (melting, sublimation, etc.). These processes, in turn, affect the flow in the vicinity of the stagnation point due to realization of the heat of phase transition, the conduction of heat from the entrained mass, and the diffusion of evaporating material into the boundary layer. References [1, 2] are devoted to a study of the joint influence of the magneto-gasdynamic and ablation effects. The magnetogasdynamic layers and the wall profile of the external velocity (flow around wedges) are discussed in [1], and special cases of such boundary layers-flow close to the stagnation line (the two-dimensional case) and close to the stagnation point (the axisymmetric case) of a blunt body are considered in [2]. Melting and evaporation are taken into account by setting the longitudinal and the transverse velocity components at the wall not equal to zero-the first taking into account the flow of the molten material and the second pyrolysis of the vapor of the surface material into the gaseous boundary layer. However, the values of these components, also the enthalpy on the wall h_w(in[1, 2] h_W 0), are not known beforehand and must be determined from the boundary conditions at the wall which express the mass and heat balances. The general formulation of the problem given in the gasdynamics case by G. A. Triskii in [3, 4], and elsewhere includes a consideration of the boundary-layer equations in the gas, the boundary-layer equations in the melted zone, and the heat conductivity equations in the solid with boundary conditions at the outer edge of the boundary layer, on the gas-molten zone interface, on the molten zone-solid interface, and inside the solid. This approach to the problem can also be utilized in the magnetogasdynamic case, as it is in this article with certain simplifying assumptions as compared with [3, 4]. In this sense, the present article is an extension of the results of [3, 4] to the field of magnetogasdynamics.In conclusion, the author thanks K. A. Lur'e for proposing the subject and useful discussions.     

MHD stagnation point flow of a micropolar fluid towards a heated surface

The problem of two dimensional stagnation point flow of an electrically conducting micropolar fluid impinging normally on a heated surface in the presence of a uniform transverse magnetic field is analyzed. The governing continuity, momentum, angular momentum, and heat equations together with the associated boundary conditions are reduced to dimensionless form using suitable similarity transformations. The reduced self similar non-linear equations are then solved numerically by an algorithm based on the finite difference discretization. The results are further refined by Richardson’s extrapolation. The effects of the magnetic parameter, the micropolar parameters, and the Prandtl number on the flow and temperature fields are predicted in tabular and graphical forms to show the important features of the solution. The study shows that the velocity and thermal boundary layers become thinner as the magnetic parameter is increased. The micropolar fluids display more reduction in shear stress as well as heat transfer rate than that exhibited by Newtonian fluids, which is beneficial in the flow and thermal control of polymeric processing.     

MHD stagnation point flow of a micropolar fluid towards a moving surface with radiation

The effects of magnetic field and thermal radiation on a micropolar fluid flow near a stagnation point towards a moving surface are investigated. Numerical solutions for the transformed non-linear ordinary differential equations are obtained using the fourth order Rung-Kutta integration scheme coupled with the shooting method. The effects of the magnetic parameter, the material parameter, the radiation parameter and the Prandtl number on the velocity, angular velocity and temperature profiles are displayed graphically and investigated. Also the numerical values of the local skin-friction coefficient and the Nusselt number for various values of the physical parameters are entered in tabular form and discussed.     

Self-similar solution of the unsteady flow in the stagnation point region of a rotating sphere with a magnetic field

The unsteady flow and heat transfer of a viscous incompressible electrically conducting fluid in the forward stagnation point region of a rotating sphere in the presence of a magnetic field are investigated in this study. The unsteadiness in the flow field is caused by the velocity at the edge of the boundary layer and the angular velocity of the rotating sphere, both varying continuously with time. The system of ordinary differential equations governing the flow is solved numerically. For some particular cases, an analytical solution is also obtained. It is found that the surface shear stresses in x- and y-directions and the surface heat transfer increase with the acceleration, the magnetic and the rotation parameters whether the magnetic field is fixed relative to the fluid or body, except that the surface shear stress in x-direction and the surface heat transfer decrease with increasing the magnetic parameter when the magnetic field is fixed relative to the body. For a certain value of the acceleration parameter, the surface shear stress in the x-direction vanishes while the surface shear stress in the y-direction and the surface heat transfer remain finite. Also, below a certain value of the acceleration parameter, reverse flow occurs in the x-component of the velocity profile. Received on 18 May 1998     

Nonequilibrium viscous flow of a multicomponent gas in the vicinity of the stagnation point of a blunt body

This paper gives the results of numerical calculations characterizing the effect of variation of the shock layer parameters on the heat transfer in the case of a multicomponent nonequilibrium-dissociating air on a wall with finite catalycity in the vicinity of the stagnation point of a spherical blunt body. Similar results for the case of a binary mixture can be found in [1–3]. It is shown that a consideration of the variation of the parameters in the nonequilibrium shock layer leads to a significant increase in heat flux to the noncatalytic wall in comparison with the theory of an asymptotically thin nonequilibrium boundary layer with equilibrium parameters on its outer boundary.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 144–147, March–April, 1971.The author thanks V. V. Lunev for useful comments in the discussion of this work.     

Calculation of convective heat flux in the vicinity of the stagnation point for partially ionized gas flow past a body

From numerical solutions of the boundary layer equations for a four-component gas mixture (E, N⁺, N₂, and N) with gas injection, approximate formulas for the heat flux as a function of the variation of λρ/c_p and h^* across the boundary layer and the magnitude of the objection are obtained (λ is the thermal conductivity of the mixture,ρ is density, c_p is the specific heat, and h^* is the enthalpy of the ideal gas state of the mixture). An effective ambipolar diffusion coefficient D^(a)(i) is introduced, making possible finite formulas for the convective heat fluxes in the “frozen” boundary layer. We study the behavior of these coefficients within the boundary layer. A formula is obtained for convective heat flux to the wall from partially ionized air for a nine-component mixture (E, O⁺, N⁺, NO⁺, O, N, NO, O₂ N₂). Even for simpler four-component gas model three effective ambipolar diffusion coefficients are necessary: $$\begin{gathered} D^{(a)} (A) = D (A, M) D^{(a)} (I) = 2D (A, M), \hfill \\ D^{(a)} (M) = [ 1 + c_e (I)] D(A, M). \hfill \\ \end{gathered} $$ Here D(A, M) is the binary diffusion coefficient of the atoms into molecules, and c_e(I) is the ion concentration at the outer edge of the boundary layer. The assumption of an infinitely large charge-exchange cross section and the other simplifying assumptions used in [1] lead to overestimation of the magnitude of the dimensionless heat flux by 7–15% for the “frozen” boundary layer case.     

MHD stagnation point flow towards heated shrinking surface subjected to heat generation/absorption

The magnetohydrodynamic(MHD) stagnation point flow of micropolar fluids towards a heated shrinking surface is analyzed.The effects of viscous dissipation and internal heat generation/absorption are taken into account.Two explicit cases,i.e.,the prescribed surface temperature(PST) and the prescribed heat flux(PHF),are discussed.The boundary layer flow and energy equations are solved by employing the homotopy analysis method.The quantities of physical interest are examined through the presentation of plots/tabulated values.It is noticed that the existence of the solutions for high shrinking parameters is associated closely with the applied magnetic field.     

MHD stagnation flow of a micropolar fluid through a porous medium

The unsteady MHD boundary layer flow of a micropolar fluid near the forward stagnation point of a two dimensional plane surface is investigated by using similarity transformations. The transformed nonlinear differential equations are solved by an analytic method, namely homotopy analysis method (HAM). The solution is valid for all values of time. The effect of MHD and porous medium, non dimensional velocity and the microrotation are presented graphically and discussed. The coefficient of skin friction is also presented graphically.     

Unsteady compressible boundary layer flow in the stagnation region of a sphere with a magnetic field

Summary An analysis is performed to study the unsteady compressible laminar boundary layer flow in the forward stagnation-point region of a sphere with a magnetic field applied normal to the surface. We have considered the case where there is an initial steady state that is perturbed by the step change in the total enthalpy at the wall. The nonlinear coupled parabolic partial differential equations governing the flow and heat transfer have been solved numerically using a finite-difference scheme. The numerical results are presented, which show the temporal development of the boundary layer. The magnetic field in the presence of variable electrical conductivity causes an overshoot in the velocity profile. Also, when the total enthalpy at the wall is suddenly increased, there is a change in the direction of transfer of heat in a small interval of time. Received 15 January 1996; accepted for publication 21 November 1996     

Numerical simulation of MHD stagnation point flow and heat transfer of a micropolar fluid towards a heated shrinking sheet

A comprehensive study of magneto hydrodynamics two‐dimensional stagnation flow with heat transfer characteristics towards a heated shrinking sheet immersed in an electrically conducting incompressible micropolar fluid in the presence of a transverse magnetic field is analyzed numerically. The governing continuity, momentum, angular momentum and heat equations together with the associated boundary conditions are first reduced to a set of self similar nonlinear ordinary differential equations using a similarity transformation and are then solved by a method based on finite difference discretization. Some important features of the flow and heat transfer in terms of normal and streamwise velocities, microrotation and temperature distributions for different values of the governing parameters are analyzed, discussed and presented through tables and graphs. The results indicate that the reverse flow caused due to shrinking of the sheet can be stopped by applying a strong magnetic field. The magnetic field enhances the shear stresses and decreases the thermal boundary layer thickness. The heat loss per unit area from the sheet decreases with an increase in the shrinking parameter. Micropolar fluids exhibit reduction in shear stresses and heat transfer rate as compared with Newtonian fluids, which may be beneficial in the flow and thermal control of polymeric processing. Copyright © 2011 John Wiley & Sons, Ltd.     

Unsteady self-similar viscous flow near a stagnation point

The problem of unsteady viscous incompressible fluid flow near a stagnation point is considered. Self-similar solutions describing plane and axisymmetric flows are constructed.     

Heat transfer in magnetohydrodynamic flow near a stagnation point

Summary Steady, axisymmetric, magnetohydrodynamic flow with a stagnation point on an infinite plane wall is considered with a magnetic field applied normal to the wall. Solutions are obtained in the form of series for the velocity, magnetic field and temperature when the magnetic field parameter () and the ratio of viscosity to magnetic diffusivity () are small. The case=O(1) is considered briefly when solutions which Meyer³⁾ obtained by physical order-of-magnitude arguments are derived mathematically as expansions in. Some remarks are made on the consistency of extending the results to flow within the boundary layer near the nose of a bluff body.     

Non-orthogonal stagnation point flow towards a stretching sheet

The steady flow of a viscous and incompressible fluid impinging at some angle of incidence on a stretching sheet is studied. It is shown that the stream function splits into a Hiemenz and a tangential component. Numerical solutions of the relevant functions as well as the structure of the flow field are presented and discussed. It is found that the free stream obliqueness is the shift of the stagnation point toward the incoming flow and it depends on the inclination angle.     

Flow in vicinity of blunt body stagnation point in diverging hypersonic stream

In the hypersonic thin shock layer approximation for a small ratio k of the densities before and after the normal shock wave the solution of [1] for the vicinity of the stagnation point of a smooth blunt body is extended to the case of nonuniform outer flow. It is shown that the effect of this nonuniformity can be taken into account with the aid of the effective shock wave radius of curvature R_*, whose introduction makes it possible to reduce to universal relations the data for different nonuniform outer flows with practically the same similarity criterion k. The results of the study are compared with numerical calculations of highly underexpanded jet flow past a sphere.Notations x, y a curvilinear coordinate system with axes directed respectively along and normal to the body surface with origin at the forward stagnation point - R radius of curvature of the meridional plane of the body surface - uV, vV., , p V ² respectively the velocity projections on the x, y axes, density, and pressure - and V freestream density and velocity The indices =0 and=1 apply to plane and axisymmetric flows Izv. AN SSSR, Mekhanika Zhidkosti i Gaza, Vol. 5, No. 3, pp. 102–105, 1970.     

Analysis of stagnation point flow toward a stretching sheet

There has been much recent interest in the stagnation point flow of a fluid toward a stretching sheet. Investigations that may include oblique stagnation flow and heat transfer to a horizontal plate all involve the same boundary value problem (BVP):

f^?+ff^″-(f^′)²+b²=0,