Influence of turbulent Prandtl number on heat transfer of a flat plate

In computations involving heat transfer in turbulent flow past bodies it is necessary to assume turbulent Prandtl number distribution across the boundary layer. A review and comparison of results obtained by different authors are given, e.g., in [1–5]. Unfortunately, the existing data are so contradictory that, at present, it does not appear to be possible to establish reliably a function that determines turbulent Prandtl number distribution across the boundary layer. The absence of sufficiently reliable and general results on the distribution of turbulent Prandtl number led to the result that in the majority of studies conducted in earlier years its value was assumed a constant and either close to or equal to one. The effect of turbulent Prandtl number on the intensity of heat transfer from a flat plate is numerically investigated in the present paper. The thermal, turbulent boundary layer equation is integrated for this purpose at different values of turbulent Prandtl number and results are compared with experimental data. Results from [6], where the thermal boundary layer was numerically integrated with Pr_t=1 and compared with experimental data, were used for comparison in the present paper. The same numerical integration procedure as in [6] was used here.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 81–85, July–August, 1984.     

Stability of a laminar boundary layer of a power-law no n-newtonian liquid

The stability of a laminar boundary layer of a power-law non-Newtonian fluid is studied. The validity of the Squire theorem on the possibility of reducing the flow stability problem for a power-law fluid relative to three-dimensional disturbances to a problem with two-dimensional disturbances is demonstrated. A numerical method of integrating the generalized Orr-Sommerfeld equation is constructed on the basis of previously proposed [1] transformations. Stability characteristics of the boundary layer on a longitudinally streamlined semiinfinite plate are considered.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 101–106, January–February, 1976.     

Supersonic boundary layer transition induced by roughness on the attachment line of a yawed cylinder

The effect of a single two-dimensional irregularity and sandy roughness on boundary layer transition in supersonic flow over a yawed cylinder (M = 6)-has been experimentally investigated. The characteristic flow regimes beyond the roughness are identified, and their limits are determined as a function of the Reynolds number and the ratio of the height of the roughness to the characteristic thickness of the boundary layer. A qualitative comparison is made with the flow regimes induced by roughness on the attachment line in incompressible flow over a cylinder [1–3]. The thermal indicator coating method is used to measure the heat fluxes along the attachment line and a comparison is made with calculations carried out in accordance with the methods of other authors.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 28–35, November–December, 1991.The authors are grateful to A. F. Kiselev for helping to calculate the heat flux in the turbulent boundary layer.     

Determination of the turbulent Prandtl number in an asymptotic boundary layer with suction

The problem of heat transfer in a turbulent asymptotic boundary layer with suction is solved in the framework of the monoharmonic model. The flow is one dimensional on the average, which is why it is chosen for investigation. The theoretically determined mean and pulsation characteristics of the flow, in particular the turbulent Prandtl number, agree with the experimental results for a boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 74–79, January–February, 1981.     

Effect of turbulence of external flow on the turbulent boundary layer at a plate

In [1, 2] turbulence of the external flow was taken into account by specifying the turbulent energy at the external boundary of the boundary layer on integrating the energy-balance equation for the turbulence. In [3] a special correction that allowed the turbulence of the external flow to be taken into account was introduced in determining the mixture path. In [4, 5] the turbulent energy calculated from the energy-balance equation of the turbulence was added to the energy induced by turbulence of the external flow, the energy distribution of the induced turbulence being specified using an empirically selected function. In [6, 7] a method of taking into account the effect of turbulence of the external flow on a layer of mixing and a jet was proposed. In the present work, this method is applied to the boundary layer at a plate.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 26–31, May–June, 1977.     

Use of the turbulence energy equation in the theory of jet flows

In this study some of the assumptions introduced in [1] in developing a closed system of equations for a turbulent boundary layer will be simplified. With the aid of the system of equations of [1], a theoretical solution is found for the problem of a jet in an accompanying flow, it being assumed that the structure of the jet turbulence depends solely on local conditions. Experiment has shown that the turbulence in such a jet does depend also on the prehistory of the flow. At large distances from the source, the theoretical characteristics of the jet agree well with the experimentally determined characteristics of the wake beyond a body. Also examined is the problem of the boundary layer between two homogeneous flows, flowing with different velocities.Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 75–81, March–April, 1973.     

Criteria for the detachment of a three-dimensional boundary layer

On the assumption that the character of the flow in a specified section of the boundary layer depends solely on the behavior of the external flow in the immediate vicinity of the section under consideration, criteria are derived for the detachment of laminar and turbulent boundary layers in the case of three-dimensional flow. The detachment criteria so found are a consequence of the necessary condition for the detachment of a three-dimensional boundary layer deduced earlier [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 50–54, July–August, 1970.     

Investigation by experiment and calculation of a supersonic stream adjacent to a wall in an accompanying supersonic flow

In well-known papers devoted to the investigation of supersonic streams adjacent to a wall, the authors, as a rule, restrict themselves to the case of a subsonic blast. In the present paper we determine the velocity field and the concentration field of an admixture of helium in a plane supersonic stream of air (M₁=2.18), propagating along a surface in an accompanying supersonic flow of air (M₂=2.7 and 3.8). In the boundary layer approximation a numerical calculation is made of the non-self-similar isobaric flow, using the equation for the turbulent viscosity [1] as the closing relationship. Results of the calculation are compared with experimental data.Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 25–32, July–August, 1972.     

Aerodynamics of end-wall boundary layer in a vortex chamber

The need for the inclusion of end-wall boundary layers in the study of the aerodynamics of vortex chambers has been frequently mentioned in the literature. However, owing to limited experimental data [1–3] with reliable information on the wall layers, the existing computational methods for end-wall boundary layers are not well-founded. The question of which parameters determine the formation of end-wall flow remains debatable. In some studies [4, 5], the vortex chambers are conditionally divided into short and long chambers. However, there is no unique opinion on the role of end-wall flows in vortex chambers of different lengths. It has also not been established for what geometric and flow parameters the chamber could be considered long or short. In the present study, as in [1, 5–8], solution is obtained for the end-wall boundary-layer equations using integral methods, considering the boundary layer in the radial direction in the form of a submerged wall jet. Such an approach made it possible to use the laws for the development of wall jets [9], and obtain fairly simple relations for integral parameters, skin friction, mass flow in the boundary layer, and other characteristics. Results are compared with available experimental data and computations of others authors; turbulent flow is considered; results for laminar boundary layer are given in [10].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 117–126, September–October, 1986.     

Design of a wing airfoil in a flow with a separated turbulent boundary layer

The problem of designing the contour of an airfoil in a viscous (incompressible and compressible) flow with a separated turbulent boundary layer from a pressure distribution given on the separationless part of the contour is solved using the boundary layer theory together with the separated flow model proposed in [1]. Numerical calculations are carried out to demonstrate the possibilities of the method.Kazan'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 83–91, May–June, 1994.     

Cone in a supersonic flow near a surface with a turbulent boundary layer

The published information about the interaction of incident shocks and a turbulent boundary layer relate to cases of a thin boundary layer ( 1–3 mm) on a flat surface. The present study relates to supersonic flow with Mach number M = 3 and stagnation pressure p₀=1.2 MPa past cones near a surface with a thick boundary layer formed on a plate abutting the lower edge of a plane nozzle.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 177–180, July–August, 1991.     

Statistical laws of the pressure fluctuations in a hypersonic turbulent boundary layer

The results of measuring the pressure fluctuations on the wall of the nozzle of a hypersonic wind tunnel beneath a developed turbulent boundary layer are presented for the Mach number M_∞ = 7.5. On the basis of a statistical analysis, it is shown that the action of the turbulent flow is dynamically similar to the propagation of a random sequence of wave packets with continuously distributed temporal and spatial scales. Low-frequency disturbances are associated with large-scale structures of long duration that propagate at a mean-statistical velocity similar in value to the outer flow velocity. The continuous generation of weakly-correlated small-scale disturbances ensuring the maintenance and development of turbulence occurs chiefly in the inner region of the boundary layer. Spectral estimates of the power generated by the turbulent flow in the wall region of the boundary layer are presented.     

Calculation of the averaged three-dimensional boundary layer at the axisymmetrical boundaries of the interblade channels of turbine machines

The article discusses the flow of a gas at the blade rim of an axial turbine, consisting of an external steady-state continuous flow of an ideal compressible liquid and a three-dimensional turbulent boundary layer of a compressible liquid at the end surfaces of the rim, averaged in a peripheral direction. It presents an example of a calculation of flow in fixed blades, with a different form of the meridional cross section. In a flow through the rim of a turbine machine between the convex and concave surfaces of adjacent blades there arises a transverse gradient of the static pressure. At the end surface in the boundary layer the lines of the flow are shifted toward the convex side of the profile, and a secondary transverse flow of the liquid arises [1–3]. The article discusses the following: an external two-dimensional steady-state adiabatic flow of an ideal compressible liquid at the surface S₂, which can be taken as the mean surface of the interblade channel, with boundary lines at the peripheral and root end surfaces of the rim; a two-dimensional steady-state adiabatic flow of an ideal compressible liquid at the end surfaces of the rim between the convex and concave sides of the profiles [3, 4]; and a three-dimensional turbulent boundary layer, averaged in a peripheral direction at the end surfaces of the blade rim. The averaged boundary layer is calculated along one coordinate line s, and a simplified model of the quasi-three-dimensional flow is used. The coefficients of friction and heat transfer, and the inclination of the bottom flow lines are averaged.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 22–31, May–June, 1975.The author thanks G. Yu. Stepanov for posing the problem and evaluating the results.     

Study of turbulent boundary layers on smooth and rough surfaces with arbitrary pressure gradients

The results of an experimental investigation into the steady-state plane turbulent boundary layer in an incompressible liquid at an impermeable wall are presented. Cases of flow at smooth and rough surfaces in the presence of a longitudinal pressure gradient are considered. The results of measurements of the turbulent structure of the flow at various distances from the channel inlet are presented. A detailed analysis of the kinematic and dynamic characteristics of the flow is given. Special attention is paid to the boundary region of the flow close to the wall. A universal law is proposed for the variation in the local resistance coefficient along the boundary layer.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 126–134, May–June, 1971.     

Experimental investigation into the reaction of a boundary layer to external periodic disturbances

The effect of a concentrated external disturbance on the boundary layer of a plate was investigated in the framework of the reaction of boundary layers to external disturbances. A disturbance localized above the surface of the plate was introduced into the external flow. Measurements revealed the generation of Tollmien—Schlichting waves in the boundary layer; in conjunction with the results of the earlier studies [1, 2], this shows that a concentrated external disturbance is an effective means of generating characteristic oscillations in a boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 155–159, July–August, 1980.     

Estimation of the acoustic emission spectrum of a boundary layer on a rough surface

Under engineering conditions the surfaces over which fluids flow are not usually hydraulically smooth. In this connection it is important to investigate the generation of sound by a turbulent boundary layer on a rough surface. Turbulent flow over a deformed surface creates dipole sources of sound, which may considerably increase the acoustic emission as compared with the quadrupole emission from a boundary layer on a smooth plate [1, 2]. In the case of sandy roughness estimates of the acoustic field are usually based on the energy summation of the fields generated by flow over the individual roughness elements [3, 4]. In this case not easily verifiable assumptions are made concerning the nature of the turbulent flow near the roughness, and the intensity of the emission is found correct to a constant factor subject to experimental determination. In the present study, in order to calculate the acoustic emission of a boundary layer on a surface with sandy roughness, it is proposed to employ the available experimental data on the cross of the surface pressure.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 20–26, September–October, 1985.     

Method of mean-mass parameters for a three-dimensional boundary layer in a flow with vorticity

When a gas flows with hypersonic velocity over a slender blunt body, the bow shock induces large entropy gradients and vorticity near the wall in the disturbed flow region (in the high-entropy layer) [1]. The boundary layer on the body develops in an essentially inhomogeneous inviscid flow, so that it is necessary to take into account the difference between the values of the gas parameters on the outer edge of the boundary layer and their values on the wall in the inviscid flow. This vortex interaction is usually accompanied by a growth in the frictional stress and heat flux at the wall [2, 3]. In three-dimensional flows in which the spreading of the gas on the windward sections of the body causes the high-entropy layer to become narrower, the vortex interaction can be expected to be particularly important. The first investigations in this direction [4–6] studied the attachment lines of a three-dimensional boundary layer. The method proposed in the present paper for calculating the heat transfer generalizes the approach realized in [5] for the attachment lines and makes it possible to take into account this effect on the complete surface of a blunt body for three-dimensional laminar, transition, or turbulent flow regime in the boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 80–87, January–February, 1981.     

Effect of nonisothermality of the surface of a thin profile on the stability of a laminar boundary layer

Investigations of the stability of a subsonic laminar boundary layer have shown that, other things being equal, the stability of the laminar flow is considerably improved by cooling the entire surface of the body to a constant temperature T_w=const lower than the temperature of the free stream [1–3]. This is attributable to an increase in the critical Reynolds number of loss of stability and a decrease in the range of unstable perturbations of the Tollmien-Schlichting wave type when the surface is cooled. Recently, in the course of investigating the stability of laminar flow over a flat plate it was found [4, 5] that a similar improvement in flow stability can be achieved by raising the temperature of a small part of the surface near the leading edge of the plate. In this study we examine the possibility of delaying the transition to turbulent flow by creating a nonuniform temperature distribution along the surface of thin profiles, where the development of an adverse pressure gradient in the outer flow has a destabilizing effect on the boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 36–42, September–October, 1986.In conclusion, the authors wish to thank M. N. Kogan for useful discussions of their results.     

The turbulent boundary layer of dissociated gas in the initial section of a tube

Many theoretical and experimental papers [1–4] have been devoted to investigating the turbulent boundary layer in the initial section of a channel. For the most part, however, the flow of an incompressible fluid with constant parameters is considered. There are many practical cases in which it is of interest to treat the development of the turbulent boundary layer of gas in the initial section of a pipe when conditions are strongly nonisothermal. A solution of a problem of this type, based on the theory of limit laws, is given in paper [1]. The present article extends this solution to the case of the flow of a high-enthalpy gas when the effect of gas dissociation on the turbulent boundary layer characteristics must be taken into account. We shall consider the flow of a mixture of i gases which is in a frozen state inside the boundary layer, and in an equilibrium state on its boundaries. Formulas are derived for the laws of friction and heat exchange, and a solution is given for the turbulent boundary layer equations in the initial section of the pipe when the wall temperature is constant and the gas flows at a subsonic velocity.Finally the authors are grateful to S. S. Kutateladze for discussing the paper.     

Boundary layer in the vicinity of the stagnation point of a body of axial symmetry in a turbulent stream

The flow in the boundary layer in the vicinity of the stagnation point of a flat plate is examined. The outer stream consists of turbulent flow of the jet type, directed normally to the plate. Assumptions concerning the connection between the pulsations in velocity and temperature in the boundary layer and the average parameters chosen on the basis of experimental data made it possible to obtain an isomorphic solution of the boundary layer equations. Equations are obtained for the friction and heat transfer at the wall in the region of gradient flow taking into account the effect of the turbulence of the impinging stream. It is shown that the friction at the wall is insensitive to the turbulence of the impinging stream, while the heat transfer is significantly increased with an increase in the pulsations of the outer flow. These properties are confirmed by the results of experimental studies [1–4].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 83–87, September–October, 1973.