Unsteady three-dimensional viscous shock layer in nonuniform gas flow in the neighborhood of a stagnation point

The combined influence of unsteady effects and free-stream nonuniformity on the variation of the flow structure near the stagnation line and the mechanical and thermal surface loads is investigated within the framework of the thin viscous shock layer model with reference to the example of the motion of blunt bodies at constant velocity through a plane temperature inhomogeneity. The dependence of the friction and heat transfer coefficients on the Reynolds number, the shape of the body and the parameters of the temperature inhomogeneity is analyzed. A number of properties of the flow are established on the basis of numerical solutions obtained over a broad range of variation of the governing parameters. By comparing the solutions obtained in the exact formulation with the calculations made in the quasisteady approximation the region of applicability of the latter is determined. In a number of cases of the motion of a body at supersonic speed in nonuniform media it is necessary to take into account the effect of the nonstationarity of the problem on the flow parameters. In particular, as the results of experiments [1] show, at Strouhal numbers of the order of unity the unsteady effects are important in the problem of the motion of a body through a temperature inhomogeneity. In a number of studies the nonstationary effect associated with supersonic motion in nonuniform media has already been investigated theoretically. In [2] the Euler equations were used, while in [3–5] the equations of a viscous shock layer were used; moreover, whereas in [3–4] the solution was limited to the neighborhood of the stagnation line, in [5] it was obtained for the entire forward surface of a sphere. The effect of free-stream nonuniformity on the structure of the viscous shock layer in steady flow past axisymmetric bodies was studied in [6, 7] and for certain particular cases of three-dimensional flow in [8–11].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 175–180, May–June, 1990.     

Hypersonic three-dimensional viscous shock layer in a nonuniform gas flow in the neighborhood of the stagnation point

Three-dimensional hypersonic viscous gas flow past smooth blunt bodies in the presence of injection or suction is considered. The effect of the nonuniformity of the approach stream on the shock-wave standoff, the flow structure and the friction and heat transfer coefficients is investigated with reference to the examples of flow from a supersonic spherical source and flow of the far wake type. It is shown that this effect depends importantly on the Reynolds number, the nature of the nonuniformity and the shape of the body.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 136–145, November–December, 1987.     

Fluid motion in the stagnation point neighborhood

An analytical solution of the inviscid incompressible fluid equations (outer flow), which serves as the asymptote for the numerical solution of the viscous-fluid equations (inner flow), is presented. Using this solution, the effect of the spreading rate and the vorticity of the outer flow on the heat transfer (heat flux "peaks") can be studied. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 97–101, July–August, 2000.     

Temperature-dependent heat sources or sinks in a stagnation point flow

Summary An analytical study has been made to determine the heat transfer characteristics of a stagnation point flow in which there are temperature-dependent heat sources or sinks. Results have been obtained for both strong and weak sources or sinks for a Prandtl number of 0.7. An analytical method, applicable to all Prandtl numbers, was utilized which circumvented the need for extensive numerical solutions and which, at the same time, provided a closed-form representation for the heat transfer. A few numerical solutions were carried out to verify the method.Nomenclature a _i constants depending on Prandtl number - c _p specific heat at constant pressure - f dimensionless velocity variable - g function defined by equation (13) - g _n functions of (n=1, 2, 3,...) - k thermal conductivity - Pr Prandtl number, c _p /k - q heat transfer rate per unit area at surface - Q heat flux parameter, q/k(u ₁/)^1/2 - S rate of heat generation or removal per unit volume (divided by c _p ) - T static temperature; T _w , wall temperature; T , free-stream temperature - u ₁ proportionality constant for free-stream velocity - U free-stream velocity - v normal velocity component - x coordinate measuring distance along surface from stagnation point - y coordinate measuring distance normal to surface - heat generation parameter, equation (3) - dimensionless normal coordinate, - dimensionless temperature - _n functions of (n=1, 2, 3,...) - absolute viscosity - kinematic viscosity - density     

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.     

Influence of the flow in the neighborhood of a stagnation point on the characteristics of an electrostatic probe

The transverse flow around an electrically conductive plate by a weakly ionized gas stream is investigated in this paper. A formulation of the problem and an algorithm of the solution are given. For certain values of the parameters the equations for the shielding and viscous layers are integrated numerically to permit estimation of the influence of the free stream velocity on the voltampere characteristic of a flat probe.     

Conditions of simulation of stagnation point heat transfer from a high-enthalpy flow

The problem of local simulation of stagnation point heat transfer to a blunt body is solved within the framework of boundary layer theory on the assumption that the simulation subsonic high-enthalpy flow is in equilibrium outside the boundary layer on the model, while the parameters of the natural flow are in equilibrium at the outer edge of the boundary layer on the body. The parameters of the simulating subsonic flow are expressed in terms of the total enthalpyH ₀, the stagnation point pressurep _w and the velocityV ₁ for the natural free-stream flow in the form of universal functions of the dimensionless modeling coefficients=R _m ^* /R _b ^* ( .<1),=V ₁/2H ₀ ( .<1) whereR _m ^* and R _b ^* are the effective radii of the model and the body at their stagnation points. Approximate conditions for modeling the heat transfer from a high-enthalpy (including hypersonic) flow to the stagnation point on a blunt body by means of hyposonic (M1) flows, corresponding to the case ²1, are obtained. The possibilities of complete local simulation of hypersonic nonequilibrium heat transfer to the stagnation point on a blunt body in the hyposonic dissociated air jets of a VGU-2 100-kilowatt induction plasma generator [4, 5] are analyzed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.1, pp. 172–180, January–February, 1993.     

Flow in the neighborhood of the stagnation point of an axisymmetric wake

The solutions of the equations of parabolic type describing the development of the flow in an axisymmetric wake under the Influence of viscosity and an adverse pressure gradient are considered. It is then shown that in the general case in the neighborhood of the stagnation point on the axis of the wake the solution is a singular one, the possibility of its continuation beyond the stagnation point being excluded. The following solutions are also obtained: a regular solution in the neighborhood of the stagnation point and a singular solution continuable downstream. This singular solution is the limit for the class of regular solutions having a miniumum in the velocity distribution on the axis as the minimum velocity tends to zero.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 53–59, March–April, 1986.The author is grateful to V. Ya. Neiland and Vik. V. Sychev for discussing the results and offering useful advice.     

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.     

Rarefied gas flow near the forward stagnation point of a blunt body at hypersonic speeds

With reduction of the density in a hypersonic stream the transition of the flow from continuum to free molecule takes place gradually. The transition region may be divided into several regimes, in each of which a definite physical phenomenon is most significant. For the case of the flow in the vicinity of the forward stagnation point of a blunt body these phenomena include increase of the thickness of the detached shock wave and of the boundary layer, the presence of viscous flow in the entire disturbed layer ahead of the blunt body, reduction of the number of collisions between molecules and the associated relaxation effects, the increasing role of the interaction of the stream molecules with the surface, and the phenomena of slip and temperature jump.     

Analytic investigation of friction and heat transfer in the neighborhood of a three-dimensional stagnation point at low and moderate Reynolds numbers

A study is made of hypersonic three-dimensional flow of a viscous gas past blunt bodies at low and moderate Reynolds numbers with allowance for the effects of slip and a jump of the temperature across the surface. The equations of the three-dimensional viscous shock layer are solved by an integral method of successive approximation and a finite-difference method in the neighborhood of the stagnation point. In the first approximation of the method an analytic solution to the problem is found. Analysis of the obtained solution leads to the proposal of a simple formula by means of which the calculation of the heat flux to a three-dimensional stagnation point is reduced to the calculation of the heat flux to an axisymmetric stagnation point. A formula for the relative heat flux obtained by generalizing Cheng's well-known formula [1] is given. The accuracy and range of applicability of the obtained expressions are estimated by comparing the analytic and numerical solutions. Three-dimensional problems of the theory of a supersonic viscous shock layer at small Reynolds numbers were considered earlier in [2–5] in a similar formulation but without allowance for the effects of slip.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 143–150, March–April, 1988.     

Effects of g-jitter and radiation on three-dimensional double diffusion stagnation point nanofluid flow

The unsteady double diffusion of the boundary layer with the nanofluid flow near a three-dimensional(3 D) stagnation point body is studied under a microgravity environment. The effects of g-jitter and thermal radiation exist under the microgravity environment, where there is a gravitational field with fluctuations. The flow problem is mathematically formulated into a system of equations derived from the physical laws and principles under the no-slip boundary condition. With the semi-similar tran...     

Stability of heat transfer regimes at the front stagnation point of a body in a stream of dissociated gas

The formulation and solution of the stationary problem of heat transfer in the neighborhood of the front point of a body at constant temperature in a stream of dissociated air are given in [1]. In [2], the results are given of numerical solution of this problem in the nonstationary formulation; the establishment of a stationary heat transfer regime was established for all the calculated variants. In the present paper, we investigate the stability of stationary heat transfer regimes at the front stagnation point of a body in a stream of dissociated air using the Lyapunov functional method [3, 4] and the method of [2, 5], which is based on the use of Meksyn's method in boundary-layer theory [6, 7]. It is established that an arbitrarily strong growth of the Damköhler number does not lead to instability and multiplicity of the stationary regimes, in contrast to the case when a hot mixture of gases flows over the front point of a thermostat [2, 5, 8]. Numerical solution of the boundary-layer equations for a wide range of Damköhler numbers confirms the results of the approximate qualitative analysis and shows that in a number of cases the time of establishment of the stationary state is a nonmonotonic function of the Damköhler number.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 97–106, September–October, 1979.     

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.     

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,     

Class of solutions for flow near a stagnation point

This paper considers a class of solutions for flow of a perfect gas near the stagnation point on a two-dimensional obstacle, where the flow is rotational far upstream from the obstacle. It is shown that the potential flow near the stagnation point is a special case of this class of solutions. Solutions accounting for the rotationality of the outer flow are obtained for flow in the mixing layer with an obstacle, and these solutions differ appreciably from the analogous Jimenez solution for potential flow near the stagnation point on a two-dimensional obstacle.