Shock waves and turbulence in a hypersonic inlet

A numerical investigation for an axisymmetric hypersonic turbulent inlet flow field of a perfect gas is presented for a three-shock configuration consisting of a biconic and a cowl. An upwind parabolized Navier-Stokes solver based on Roe's scheme is used to compute an oncoming flow Mach numberM =8, temperatureT =216 K, and pressureP =5.5293×10³ N/m². In order to assess the flow quantities, the interaction between shock and turbulence, and the inlet efficiency, three different flow calculations — laminar, turbulent with incompressible and compressible two-equationk- turbulence models — have been performed in this work.Computational results show that turbulence is markedly enhanced across an oblique shock with step-like increases in turbulence kinetic energy and dissipation rate. This enhancement is at the expense of the mean kinetic energy of the flow. Therefore, the velocity behind the shock is smaller in turbulent flow and hence the shock becomes stronger. The entropy increase through a shock is caused not only by the amplification of random molecular motion, but also by the enhancement of the chaotic turbulent flow motion. However, only the compressiblek- turbulence model can properly predict a decrease in turbulence length scale across a shock. Our numerical simulation reveals that the incompressiblek- turbulence model exaggerates the interaction between shock and turbulence with turbulence kinetic energy and dissipation rate remaining high and almost undissipated far beyond the shock region. It is shown that proper modeling of turbulence is essential for a realistic prediction of hypersonic inlet flowfield. The performed study shows that the viscous effect is not restricted in the boundary layer but extends into the main flow behind a shock wave. The loss of the available energy in the inlet performance therefore needs to be determined from the shock-turbulence interaction. The present study predicts that the inlet efficiency becomes relatively lower when turbulence is taken into account.     

Unsteady flow past a cascade of staggered thin profiles

Many studies have been made of plane flow of an incompressible inviscid fluid past a cascade of profiles with arbitrary stagger angle 0. For example, in the particular case of the motion of a cascade with the stagger angle at zero-oscillation phase-shift angle =0 Khaskind [1] determined the unsteady lift force theoretically by isolating the singularities with the Sedov method [2], applying a conformal mapping to the cascade of unstaggered flat plates. Belotserkovskii et al. [3] calculated the over-all unsteady aerodynamic characteristics of a cascade in the particular case =0 and for any on a computer by the method of discrete vortices, and for the more general case (0) Whitehead [4] has done the same using a vortex method. Gorelov and Dominas [5] calculated the over-all unsteady force and moment coefficients of a profile in a cascade with stagger angle 0 and phase shift 0.The calculation method was based on unsteady theory for a slender isolated profile whose flow pattern is known, with subsequent account for the interference of the profiles and the vortex wakes behind them.In the present study the singularity isolation method [2] is extended to slender profiles with arbitrary stagger angle 0 and arbitrary phase shift 0 of the oscillations between neighboring profiles. It is shown that the solution reduces to the solution of a Fredholm integral equation of the first kind in terms of the sum of the tangential velocity components along the profile. It is found that the relative effect of the unsteady flow due to the system of vortex trails behind the cascade with stagger may be determined without solving this integral equation. However, this solution must be found to calculate the added masses of the cascade and the total magnitudes of the unsteady forces. It is found that regularization transforms the integral equation of the first kind to an integral equation of the second kind, for which solution methods are known.Thus the expressions for the unsteady forces are determined in the form of separate terms, each of which has a physical significance: as a result we obtain finite formulas (improper integrals) for calculating the variable forces; from these formulas are derived the asymptotic expressions for the forces in the limiting cases of high and low solidities and Strouhal numbers, which as a rule are lost in numerical calculations. The proposed method may be considered as one of the techniques for improving the convergence of the numerical methods (elimination of singularities). Moreover, this method may be used to solve the problems of unsteady flow past cascades of arbitrary systems of slender profiles for various profile incidence angles relative to the x-axis and in the presence of a finite cavitation zone on the profiles.The limited practical application of this method is explained by the extreme theoretical difficulties in its applications to cascades with stagger angle. In the present studies these difficulties are examined using the example of a cascade with stagger angle.     

Effect of Strong External Turbulence on a Wall Jet Boundary Layer

The initial stage of the development of a wall jet under the influence of strong external turbulence has been studied in a novel shear-flow mixing-box experiment. A fully developed channel flow of depth h (40 mm) enters along the top wall of a cuboidal box of height 11 h in which a combination of oscillatory and turbulent velocity fluctuations are generated by a vertical oscillating grid at the midplane 5 h below the wall. When the ratio of the rms grid-generated velocity fluctuations, , to the local mean velocity inside the wall jet layer, u, is greater than about 0.1, significant changes are observed in the mean shear profile and in the eddy structure of the wall jet. The wall jet thickness increases by approximately 25% but the maximum velocity decreases by less than 10% compared to the case without the external turbulence. Fluctuations of the streamwise velocity component increase as expected in the outer part of the wall jet, but the most significant result is the increase by 70% of the fluctuations in the boundary layer close to the wall. CFD simulations using the k-ɛ RNG of the FLUENT CFD Code do not properly model the effect of the large scale external turbulence in this experiment. However, an artificial method, which introduces a series of small inlet/outlet jets to represent external turbulence, approximately simulates the overall effects of the oscillating grid on the wall jet, but does not simulate the amplification of the near wall turbulence. F. T. M. Nieuwstadt: Rest in peace (1946–2005).     

Effect of Free-Stream Turbulence on the Reduction of Surface Friction in a Turbulent Boundary Layer Behind LEBU-Devices

The effect of increased free-stream turbulence on the reduction of the surface friction coefficient c _f in a turbulent boundary layer behind large-eddy break-up (LEBU) devices is investigated using a gravimetric method. The turbulence level was ε ≈ 1.9–4.9 % and the turbulence scale L _e ≈ 40–110 mm. The boundary layer Reynolds number Re** was varied from 2300 to 7500, with the boundary layer thickness being varied on the range δ = 33–44 mm. It is shown that an increase in the turbulence level ε has almost no impact on the relative reduction of friction behind LEBU-devices, whereas, under similar conditions of elevated free-stream turbulence, for another method, namely, the use of surface riblets, the friction reduction may be more strongly expressed.     

Experimental investigations on steady wake effects in a high-lift turbine cascade

This paper reports on the investigation of steady wake effects in cascades. An annular cascade rig, where two stators having the same blade pitch can be circumferentially traversed relatively to each other, is used to analyse the profile losses and the boundary layer development of the downstream stator for different circumferential positions of the upstream stator (clocking positions). Different measurement techniques are used such as three-hole pressure probes, and hot wire- and surface-mounted hot-film probes. The results show a varying pressure loss coefficient of the downstream cascade (S2) for different clocking positions of the upstream cascade (S1_SP).     

Flow Structure of Lifted Swirling Jet Flames

Swirling combustion is widely applied in various applications such as gas turbines, utility boilersor waste incinerators. This article contributes to the ongoing research by providing experimentaldata that are gathered in the mixing zone of a lifted swirling premixed natural gas flame. Theobjective of this paper is fivefold: (1) to introduce the lifted swirling flame featuring lowNO _x emissions (2) to provide experimental data such as major species distributions, temperature and streamlines of the flow pattern, (3) to report on velocity bias in probability density function (PDF) distributions and to present PDF sequences of velocities in medium scale swirling flows, (4) to make an assessment on the local small-scale turbulence that is present in the swirling mixinglayer and (5) to provide new experimental data for model verification and development.The PDFs are corrected in order to compensate for the velocity bias phenomenon, which is typicalfor randomly sampled LDA data. Sequences of axial PDF data are presented and measurement locationsof interest are selected to look at the PDF characteristics of the internal and externalrecirculation zones, the mixing layer and the onset of the reacting flow into detail. The mixinglayer PDFs covered a wide velocity range and revealed bimodality; even the concept ofmulti-modality is suggested and explored. Analysis showed that a sum of two Gaussian distributionscan accurately envelop the experimental PDFs. The reason for this broadband turbulence behavior isto be found in combination of precessing and flapping motion of the flow structures, and also incombustion generated instabilities of the lifted flame. As a result, the flame brush is wide (largescale motion) and the mixing (small-scale turbulence) flattens any high temperatures in thecombustion process.The multi-scale turbulence concept is subsequently used to make anassessment of the local turbulence characteristics in the mixing layer.The idea is that the PDFs capture both contributions of the flow-inherent fine grain turbulence (u _l ) which is superposed on slowlarge scale fluctuating structures. It is this u _l that will be of interest in continued research on the classification of the lifted flame into acombustion regime diagram (e.g. Borghi diagram). Finally, the bimodalitycharacter in reacting flows and the prediction of large-scale structuresmay be a challenge for LES researchers.     

Numerical Study of Hydrodynamics and Heat and Mass Transfer of a Ducted Gas–Vapor‐Droplet Flow

A computational model has been developed to predict heat and mass transfer and hydrodynamic characteristics of a turbulent gas–vapor–droplet flow. Turbulent characteristics of the gas phase are computed using the k– model of turbulence. It is shown that, with increasing inlet droplet diameter, the rate of heat transfer between the duct surface and the vapor–gas mixture decreases appreciably, whereas the wall friction increases only insignificantly. The predicted values agree fairly well with available experimental and numerical data     

Effect of angle of attack on supersonic flow past axisymmetric blunt bodies with surface injection

The effect of angles of attack in the interval 0 40° on the flow pattern and the aerodynamic characteristics of a body of power-law shape (equation of the generator in the cylindrical coordinate system r=zⁿ, n=0.125) is investigated for supersonic flow without injection and with intense subsonic localized injection from the surface. As a result of numerical calculations it is established that the use of Newton's theory for determining the coordinates of the gas stagnation point behind the shock in flow past an impermeable body of the shape in question leads to serious errors, and an expression for determining the location of this point is given. It is shown that for three-dimensional flow the flow pattern and the surface pressure distribution are sharply different from the case=0. It is established that on the parameter interval in question intense injection considerably reduces the aerodynamic drag without loss of static stability, which is important in connection with the solution of the problem of gas-stable aircraft control.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 95–101, September–October, 1987.     

Steady flow around a plane cascade by an ideal gas

The steady separation-free flow around a flat cascade by an ideal gas is discussed. Most of the attention is devoted to blocking regimes with a supersonic velocity in the entire flow and its subsonic component normal to the front of the cascade. A directing action of the cascade (the direction of the velocity and the Mach number of the advancing flow turn out to be related) is exhibited in these regimes which is a consequence of an independence of the flow in front of the cascade of the conditions behind it [1–5]. The most widespread method of their calculation [3, 4, 6] is based on the method of characteristics with establishment of the flow outside the cascade in a timelike coordinate. Although the integrated conservation laws also permit finding the parameters at infinity, the numerical construction of as long-range fields as desired with periodic sequences of attenuating discontinuities is practically impossible. The approximation of nonlinear acoustics (ANA) [7, 8] is justified here, as it is very effective in such problems [8–12]. A combination of ANA, the integrated conservation laws, and establishment in a calculation according to [13, 14] with isolation of the discontinuities has been realized in [5] for the construction of a solution on the entrance section of a cascade and everywhere in front of it. Below the method of [5] is extended to the entire flow and simplified even more. The flow on the entrance section of the cascade is, just as in [3], found in the approximation of a simple wave, in the rest of it and in a finite strip behind it-the flow is found with the help of the straight-through version of the scheme of [13, 14], and in the long-range field-in the ANA. A simpler version is proposed. In it ANA is applied outside the cascade and the linear theory is applied inside the cascade. Examples of the calculations are given. Similarity laws are formulated for all the regimes of streamline flow.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 35–43, November–December, 1984.     

Flow of a supersonic ideal gas over a planar cascade in the case of a subsonic normal component in regimes with attached shocks

The supersonic flow of an inviscid gas that does not conduct heat over a cascade of planar pointed profiles is considered in the case when the component of the velocity vector of the undisturbed flow normal to the cascade front is subsonic. The investigation is restricted to regimes without separation and shock waves attached to the leading edges of the profiles and fairly dense cascades, for which the characteristics or shock waves leaving the trailing edges do not enter the region in front of the cascade. In such cases, the conditions behind the cascade do not influence the flow in front of it. In this sense, the flow in the cascade, as in a Laval nozzle in the case of supercritical gradients is trapped, In the hodograph plane, trapped regimes of flow over the cascade correspond to velocity vectors of the undisturbed flow that lie on a certain line (see, for example, [1–3]), which is constructed in the process of solution of the problem. This property has been called the directing influence of the cascade on the oncoming flow. Regimes with detached shocks can also be trapped if the separation of the shocks is due to the profiles being blunt. A method is proposed that for regimes with attached shocks makes it possible to calculate the entire flow field, including the wave structure at large distances from the cascade front; some results obtained by the method are also given. The study of regimes with attached shocks, for which the analysis is simplest, is, first, of interest in its own right and, second, is a stage in the creation of methods of calculation and subsequent investigation of cascades with arbitrary regimes.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 108–113, July–August, 1979.We are grateful to M. Ya. Ivanov for assistance in updating the supersonic flow calculation program of [7], to G. Yu. Stepanov for helpful comments, and to E. V. Buganov and V. A. Vostretsov for assistance in preparing the paper.     

Temperature separation and friction losses in vortex tube

The process of energy separation and friction losses in a vortex tube is studied in detail. The hot and cold exit air temperatures were measured. Experiments have been conducted at inlet pressure of 3.5, 5, 7.5 and 9 bar, at inlet temperature of 292.15 and 298.15 K and at cold air mass ratio from 0 to1. The results demonstrate that the hot air temperature reaches its maximum value at a cold air mass ratio of nearly 0.82, while the minimum value of cold air temperature is found at a cold air mass ratio of 0.3. Based on energy and mass balances as well as on the definition of internal energy and on experimental results a new model for the determination of hot and cold exit gas temperature has been developed. The model includes the relevant primary parameters and predicts the experimental results as well as the data published in the literature sufficiently accurate for engineering purposes.A cross-section area m³ - D diameter of the pipe m - F model parameter - f friction factor - L length of the tube m - m mass flow rate kg/s - y cold air mass ratio - P static pressure Pa - T temperature K - t thickness of the orifice m - R gas constant J/kg K - v velocity of fluid m/s - density of the fluid kg/m³ - friction factor for pipe - friction factor for orifice and tee junction - 1 inlet of compressed gas - 2 exit of hot gas - 3 exit of cold gas - atm atmospheric pressure - c cold exit gas - f friction - h hot exit gas - o orifice plate - T tee junction     

Heat transfer enhancement with deformation of the velocity profile

The problem of enhancing the heat transfer in channels and boundary layers by the appropriate deformation of the fluid velocity profile is considered. The resulting additional hydraulic losses, the price of heat transfer enhancement, are determined. The possibilities of controlling heat transfer by redistributing the fluid velocity in channels are demonstrated with reference to flows at low Prandtl numbers. Laminar and turbulent liquid and gas flows with heat transfer in channels and boundary layers are numerically modeled on the basis of modern models of turbulence (flow development in channels with different initial velocity profiles, flows with wall roughness and boundary layer flows with forces acting on the flow to cause deformation of the velocity profile). In all cases it is found that the heat transfer can be enhanced only at the expense of a considerable increase in the hydaulic losses. A class of self-similar thermal problems for flows in plane diffusers is formulated. The eigenfunctions — temperature modes — for various velocity profiles are determined with allowance for the nonuniqueness of the solution of the classical dynamical problem for a plane diffuser and the corresponding heat transfer coefficients are found.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 94–105, May–June, 1993.The authors are grateful to A. Yu. Klimenko for useful discussions.     

Direct drag measurements in a turbulent flat-plate boundary layer with turbulence manipulators

The effect of turbulence manipulators on the turbulent boundary layer above a flat plate has been investigated. These turbulence manipulators are often referred to as Large Eddy Break Up (LEBU) devices. The basic idea is that thin blades or airfoils are inserted into the turbulent flow in order to reduce the fluctuating vertical velocity component v above the flat plate. In this way, the turbulent momentum transfer and with it the wall shear stress downstream of the manipulator should be decreased. In our experiments, for comparison, a merely drag-producing wire also was inserted into the boundary layer.In particular, the trade-off between the drag of the turbulence manipulator and the drag reduction due to the shear-stress reduction on the flat plate downstream of the manipulator has been considered. The measurements were carried out with very accurate force balances for both the manipulator drag and the shear stress on the flat plate. As it turns out, no net drag reduction is found for a fairly large set of configurations. A single thin blade as a manipulator performed best, i.e., it was closest to break-even. However, a further improvement is unlikely, because the device drag of the thin blade elements used here has already been reduced to only that due to laminar skin friction, and is thus the minimum possible drag. Airfoils performed slightly worse, because their device drag was higher. A purely drag-producing wire device performed disastrously. The wire device, which consisted of a wire with another thin wire wound around it to suppress coherent vortex shedding and vibration, was designed to have (and did have) the same drag as the airfoil manipulator with which it was compared. The comparison showed that airfoil and blade manipulators recovered 75–90% of their device drag through a shear-stress reduction downstream, whereas the wire device recovered only about 25–30% of its device drag.Conventional LEBU manipulators with airfoils or thin blades produce between 0.25% and 1% net drag increase, whereas the wire device (with equal device drag) produces as much as 4% net drag increase. These data are valid for the specific plate length of our experiments, which was long enough in downstream extent to realize the full effect of the LEBU manipulators. Turbulence manipulators do indeed decrease the turbulent momentum exchange in the boundary layer by rectifying the turbulent fluctuations. This generates a significant shear-stress reduction downstream, which is much more than just the effect of the wake of the manipulator. However, the device drag of the manipulator cannot be reduced without simultaneously reducing the skin friction reduction. Thus, the manipulator's device drag exceeds, or at best cancels, the drag reduction achieved by the shear-stress reduction downstream. A critical survey of previous investigations shows that the suggestion that turbulence manipulators may produce net drag reduction is also not supported by the available previous drag force measurements. The issue had been stirred up by less conclusive measurements based on local velocity data, i.e., data collected using the so-called momentum balance technique.List of symbols b lateral breadth of test plate - c chord length of turbulence manipulator - d diameter of wire manipulator - e distance of the elastic center from the leading edge of the manipulator airfoil - h height of manipulator above test plate - q dynamic pressure of the potential flow above the test plate - s spacing of turbulence manipulator elements - t thickness of turbulence manipulator elements - u,v,w fluctuating velocities in downstream, platenormal, and lateral directions - x distance from the leading edge of the test plate in the downstream direction - x ₀ location of the trailing edge of the first manipulator - z distance from test plate center in the lateral direction - C _D drag coefficient - C _L lift coefficient - D _m drag of manipulated plate including device drag and shear stress, calculated from manipulator location to downstream location - D ₀ drag of unmanipulated plate boundary layer, consisting of the shear stress calculated from manipulator location to downstream location - F drag force - F ₀ total skin friction force, measured over a distance from 0.4 m upstream of manipulator to 6.35 m downstream of manipulator, measured without turbulence manipulator - F _LEBU device drag force of the LEBU, i.e., the turbulence manipulator - F _m total drag force of manipulated plate, consisting of - F _LEBU and skin friction force, measured over a distance from 0.4 m upstream of manipulator to 6.35 m downstream - F _cf skin friction force as measured by the floating element balance, manipulated case - F _cfo skin friction force, as measured by the floating element balance, unmanipulated case - F _cf skin friction saving, defined as F _cf = F _cf – F _cfo - F _cf cumulative skin friction savings, i.e., the sum of the skin friction savings F _cf , added up from the location of the manipulator to the downstream location , as shown in Fig. 11. In Fig. 13 the cumulative skin friction savings are summarized up to their asymptotic value, reached at 200 - Re _c Reynolds number of the manipulator elements, calculated with the chord length c and the local velocity in the boundary layer - Re ₀ Reynolds number at the location x ₀ of the manipulator, calculated with the momentum thickness of the boundary layer and the mean flow velocity U - U mean flow velocity in the potential regime of the wind tunnel test section - angle of attack of the manipulator airfoils - ₀ boundary layer thickness at the location x ₀ of the manipulator - dimensionless distance from the manipulator in the downstream direction, defined as - density of the air - ₀ local skin friction shear stress, unmanipulated case - _{0 Average} skin friction shear stress, average value over the lateral span (b = 2 m) of the test plate, unmanipulated case - _m local skin friction shear stress, manipulated case - momentum thickness of the undisturbed turbulent boundary layer at the location x ₀ The authors would like to thank Prof. H. H. Fernholz for his scientific and administrative support. The hardware for the experiments was designed and built by C. Daase, W. Hage and R. Makris. Funding for the project was provided by the Deutsche Forschungsgemeinschaft and is gratefully acknowledged.     

Three-dimensional thin viscous shock layer in the absence of planes of symmetry in the flow

The basic laws of viscous homogeneous gas flow at high supersonic speeds past smooth blunt bodies with a permeable surface are investigated within the framework of the thin viscous shock layer model. An efficient numerical method of solving these equations, which makes it possible to consider cases of flow past bodies at angles of attack and slip, when there are no planes of symmetry in the flow, is proposed. Some results of calculating the flow past a triaxial ellipsoid with an axial ratio of 103n73 at angles of attack =0–45° and slip angles =0–45° over a broad interval of Reynolds numbers are presented as an example. The effect of the principal determining parameters of the problem on the flow structure in the shock layer and the surface friction and heat transfer coefficients is analyzed. An expression for calculating the heat fluxes to the impermeable surface of smooth blunt bodies in a supersonic homogeneous viscous gas flow over a broad interval of Reynolds numbers is proposed on the basis of the solutions obtained and the results of other authors.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 150–158, March–April, 1989.     

Prediction of mono-disperse gas–liquid turbulent flow in a vertical pipe

A two-fluid model in the Eulerian–Eulerian framework has been implemented for the prediction of gas volume fraction, mean phasic velocities, and the liquid phase turbulence properties for gas–liquid upward flow in a vertical pipe. The governing two-fluid transport equations are discretized using the finite volume method and a low Reynolds number $k-\varepsilon$ model is used to predict the turbulence field for the continuous liquid phase. In the present analysis, a fully developed one-dimensional flow is considered where the gas volume fraction profile is predicted using the radial force balance for the bubble phase. The current study investigates: (1) the turbulence modulation terms which represent the effect of bubbles on the liquid phase turbulence in the k–ε transport equations; (2) the role of the bubble induced turbulent viscosity compared to turbulence generated by shear; and (3) the effect of bubble size on the radial forces which results in either a center-peak or a wall-peak in the gas volume fraction profiles. The results obtained from the current simulation are generally in good agreement with the experimental data, and somewhat improved over the predictions of some previous numerical studies.     

Analysis of turbulent boundary layer interaction with an external supersonic flow behind a step

An integral method of analyzing turbulent flow behind plane and axisymmetric steps is proposed, which will permit calculation of the pressure distribution, the displacement thickness, the momentum-loss thickness, and the friction in the zone of boundary layer interaction with an external ideal flow. The characteristics of an incompressible turbulent equilibrium boundary layer are used to analyze the flow behind the step, and the parameters of the compressible boundary layer flow are connected with the parameters of the incompressible boundary layer flow by using the Cowles-Crocco transformation.A large number of theoretical and experimental papers devoted to this topic can be mentioned. Let us consider just two [1, 2], which are similar to the method proposed herein, wherein the parameter distribution of the flow of a plane nearby turbulent wake is analyzed. The flow behind the body in these papers is separated into a zone of isobaric flow and a zone of boundary layer interaction with an external ideal flow. The jet boundary layer in the interaction zone is analyzed by the method of integral relations.The flow behind plane and axisymmetric steps is analyzed on the basis of a scheme of boundary layer interaction with an external ideal supersonic stream. The results of the analysis by the method proposed are compared with known experimental data.Notation x, y longitudinal and transverse coordinates - X, Y transformed longitudinal and transverse coordinates - , ^*, ^** boundary layer thickness, displacement thickness, momentum-loss thickness of a boundary layer - , ^*, ^** layer thickness, displacement thickness, momentum-loss thickness of an incompressible boundary layer - u, velocity and density of a compressible boundary layer - U, velocity and density of the incompressible boundary layer - , stream function of the compressible and incompressible boundary layers - , dynamic coefficient of viscosity of the compressible and incompressible boundary layers - r₁ radius of the base part of an axisymmetric body - r radius - R transformed radius - M Mach number - friction stress - p pressure - a speed of sound - s enthalpy - v Prandtl-Mayer angle - P Prandtl number - P_t turbulent Prandtl number - r₂ radius of the base sting - b step depth - =0 for plane flow - =1 for axisymmetric flow Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 33–40, May–June, 1971.In conclusion, the authors are grateful to M. Ya. Yudelovich and E. N. Bondarev for useful comments and discussions.     

Velocity characteristics of the flow around a square cross section cylinder placed near a channel wall

Laser-Doppler measurements are reported of the flow around a square cross section cylinder placed at various heights (Y ₀) above a plane channel wall for a Reynolds number Re _H = 1.36 × 10⁴.The thickness of the turbulent boundary layer on the channel wall at the obstacle position, but with it removed from the water tunnel, was equal to 0.8 H, being H the square obstacle height and the free stream turbulence intensity was 6%. The periodic character of the flow in the near wake was characterized by measurements of turbulence spectra in the range 0 Y ₀/H 3.3 and the results revealed that regular vortex shedding was suppressed for a gap height less than 0.35 H. Detailed results of time averaged mean flow properties, turbulence intensities and Reynolds stresses revealed the structural differences of the near wakes with and without vortex shedding for Y _O = 0.5 and Y _O = 0.25 respectively.List of symbols R _e Reynolds number Re = U ₀ H/v - H Square obstacle height - Y Normal coordinate - Y ₀ Gap distance between the plane wall and obstacle face - C _f Skin friction coefficient - k Turbulent kinetic energy - u Friction velocity - Turbulent boundary layer thickness - y ⁺ Non-dimensional log law coordinate - X _r Separated flow length behind the obstacle - U Mean axial velocity - V Mean radial velocity - u² Axial turbulent stress - ² Normal turbulent stress - u Turbulent shear stress - U ₀ Mean bulk velocity - E Power spectrum energy - f Predominant frequency - h Distance between inner shear layers behind the obstacle - s Distance from the wall to the shear layer behind the obstacle - C _D Drag coefficient     

Control of the aerodynamic characteristics of a profile oscillating with respect to the incidence angle in subsonic viscous flow

The results of a numerical simulation of the unsteady subsonic viscous gas flow around a two-dimensional profile oscillating with respect to the incidence angle are presented and the possibility of controlling the nonstationary aerodynamic characteristics is considered. The hysteresis phenomena typical of oscillatory profile motions are investigated, the dependence of the lift force and drag is found for various laws of periodic variation of the incidence angle with time, and the effect of the frequency and amplitude of the angular profile oscillations on the shape of the hysteresis curves is studied. The calculations were based on the numerical solution of the nonstationary Navier-Stokes equations averaged in the Reynolds sense (Reynolds equations) which were closed using the k-ω turbulence model with modeling of the laminar/turbulent transition.     

Effect of free-stream turbulence on surface friction in a turbulent boundary layer

The combined effect of the turbulence intensity , the turbulence scaleL, and the Reynolds number Re^** on the surface friction coefficientc _f in a turbulent boundary layer is studied. The dependence of the relative friction increment on the equivalent turbulence level _cq, which takes into account the simultaneous variation in ,L and Re^**, is determined. The threshold value _cq ^* below which the value ofc _f does not depend on _cq is found.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 65–75, March–April, 1995.     

Dfag measurements on star-shaped body at m ≈ 6 and 8

Results have been obtained in recent years which make it possible to get an idea of the optimal shape of a three-dimensional body at high supersonic speeds. It has been shown [1–6] that bodies with a cross section in the form of a star with certain limitations have the least wave drag and remain optimal with respect to total drag with approximate account for the friction forces. The transition from the optimal body of revolution to the star-shaped body of equivalent volume and length makes a several-fold drag reduction. These theoretical results, initially obtained on the basis of the Newton drag law, were then confirmed by the exact solution [7] for bodies which were close in form to the optimal. Subsequent experimental studies investigated the flow pattern between two lobes representing an element of the star over a wide range of included angles. The experiments showed that there actually exists a flow between the rays corresponding to the solution [7], that this flow is stable, and that the wave drag calculated from the pressure distribution over the body surface is several fold less than for the equivalent cone. Although these results are encouraging, they do not prove the advantages of the star-shaped form for practical use. The point is that the star has considerably more wetted area; therefore the effect of the marked reduction of the wave drag may be compensated by an increase of the friction drag. The references above to the theory which considers friction are not convincing, since the friction estimates are approximate, while real friction is complicated by the presence of shock waves within the flow, the possibility of a turbulent boundary layer, separation, etc. Not all these factors are amenable to calculation, and it is clear that conclusions can be drawn on star drag only after making direct measurements of the total force acting on a model in a flow.In the following we describe the results of force tests conducted with a star model at M6 and 8. During the tests the flow pattern in the wake behind the body was photographed in addition to the force measurements.The authors wish to thank G. I. Petrov, G. G. Chernyi, M. Ya. Yudelovich, and A. A. Churilin for assistance in carrying out the experimentation.