Wave processes in the shock layer on a flat plate at an angle of attack

A numerical and experimental study of receptivity of the viscous shock layer on a flat plate aligned at an angle of attack to external acoustic perturbations is performed. Density and pressure fluctuations are measured in experiments at the free-stream Mach number M _∞ = 21 and Reynolds number Re ₁ = 6·10 ⁵ m ⁻¹ . Direct numerical simulations of receptivity of the viscous shock layer to external acoustic perturbations in wide ranges of the governing parameters are performed by solving the Navier-Stokes equations with the use of high-order shock-capturing schemes. The calculated intensities of density and pressure fluctuations are found to be in good agreement with experimental data. Results of the study show that entropy-vortex disturbances dominate in the shock layer at small angles of attack, whereas acoustic perturbations prevail at angles of attack above 20°.     

Numerical simulation of receptivity of a hypersonic boundary layer to acoustic disturbances

Direct numerical simulations of the evolution of disturbances in a viscous shock layer on a flat plate are performed for a free-stream Mach number M _∞ = 21 and Reynolds number Re _L = 1.44 · 10⁵. Unsteady Navier-Stokes equations are solved by a high-order shock-capturing scheme. Processes of receptivity and instability development in a shock layer excited by external acoustic waves are considered. Direct numerical simulations are demonstrated to agree well with results obtained by the locally parallel linear stability theory (with allowance for the shock-wave effect) and with experimental measurements in a hypersonic wind tunnel. Mechanisms of conversion of external disturbances to instability waves in a hypersonic shock layer are discussed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 84–91, May–June, 2007.     

Bispectral analysis of numerical simulations of wave processes in hypersonic shock layers

Results of numerical simulations of the evolution of disturbances in a hypersonic shock layer on a flat plate at high Mach numbers (M_∞ = 21) and moderate Reynolds numbers (Re _L = 1.44 · 10⁵) are analyzed by an adapted method of bispectral analysis. All basic types of nonlinear interactions are obtained. The calculated results are compared with experimental data.     

Experimental study and direct numerical simulation of the evolution of disturbances in a viscous shock layer on a flat plate

The evolution of disturbances in a hypersonic viscous shock layer on a flat plate excited by slow-mode acoustic waves is considered numerically and experimentally. The parameters measured in the experiments performed with a free-stream Mach number M _∞ = 21 and Reynolds number Re _L = 1.44 · 10⁵ are the transverse profiles of the mean density and Mach number, the spectra of density fluctuations, and growth rates of natural disturbances. Direct numerical simulation of propagation of disturbances is performed by solving the Navier-Stokes equations with a high-order shock-capturing scheme. The numerical and experimental data characterizing the mean flow field, intensity of density fluctuations, and their growth rates are found to be in good agreement. Possible mechanisms of disturbance generation and evolution in the shock layer at hypersonic velocities are discussed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 5, pp. 3–15, September–October, 2006.     

Experimental study of vortex structures in a hypersonic shock layer on a flat plate

The characteristics of travelling perturbations of density in a hypersonic shock layer on a flat plate for the Mach number M_∞=21 and unit Reynolds numberRe ₁=6·10⁵ m⁻¹ were experimentally studied by the method of electron-beam fluorescence. The perturbations were generated by interaction of the shock layer behind an oblique gas-dynamic whistle and the leading edge of the plate. The cases of unsteady and quasi-steady interaction were considered. In both cases, vortex disturbances of finite amplitude were generated. The measurements were performed at the fundamental frequency F=0.6·10⁻⁴ and at the harmonic; the streamwise phase velocities, the growth rates of the disturbances, and the angles of wave propagation were obtained. The measurement results are compared with some experimental data for subsonic flows, some particular results of the linear stability theory for compressible flows, and the results obtained on the basis of a simple model of the nonlinear stage of disturbance evolution in a hypersonic boundary layer. Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 6, pp. 41–47, November–December, 1999.     

Correlation of the separation region length in shock wave/channel boundary layer interaction

 Results of an experimental investigation of the characteristics of a separation region induced by the interaction of an externally generated oblique shock with the turbulent boundary layer formed in a rectangular half channel are discussed. The experiments were carried out in the supersonic wind tunnel of the Institute of Theoretical and Applied Mechanics SB RAS at a free-stream Mach number M _∞=3.01 over a range of Reynolds numbers Re ₁=(9.7–47.5)×10⁶ m^-1 and at zero incidence and zero yaw of the model. Particular attention is paid to the size of the zone of the upstream propagation of disturbances (upstream influence region) under different experimental conditions: with varied values of the shock wave strength, half channel width, and Reynolds number. It is shown, in particular, that the normalized upstream influence region length as a function of inclination angle of the shock generator in a rectangular half channel is readily approximated by a simple exponential function. In support of the known reference data obtained for supersonic numbers M _∞ and moderate Re in other configurations, it is also shown that the upstream influence region length decreases with increasing Reynolds number. Generalization of experimental data on the length of the upstream influence region formed in similar geometric configurations is possible using an additional reference linear scale which is the distance from the leading edge of the shock generator to the exposed surface. A substantial dependence of the reference dimensions of separation region on the half channel width is also established. Received: 20 January 1997/Accepted: 30 June 1997     

Mach reflection of a plane shock wave passing a mountain of 45° inclination

The transition from regular reflection (RR) to Mach reflection (MR) as a plane shock wave diffracts around a triangular mountain of 45° inclination is analysed in this paper, both by optical measurement in a shock tube and by numerical simulation the numerical method developed by Li Yingfan^[1] is of the FLIC type with triangular mesh. The dependence of the critical transition point Lk ofRR→MR on shock Mach numberM _i is analyzed and the variations of the incidence angle ω _i of the impinging shock and the reflection angle ω _r with the distanceL ^* are investigated. Our experimental and numerical results agree well with the theoretical results of Iton and Italya.     

Two-dimensional interaction between an incident shock and a turbulent boundary layer in the presence of an entropy layer

The results of an experimental and numerical investigation of flow and heat transfer in the region of the interaction between an incident oblique shock and turbulent boundary layers on sharp and blunt plates are presented for the Mach numbers M_∞ = 5 and 6 and the Reynolds numbers Re_∞L = 27×10⁶ and 14×10⁶. The plate bluntness and the incident shock position were varied. It is shown that the maximum Stanton number St _m in the shock incidence zone decreases with increase in the plate bluntness radius r to a certain value and then varies only slightly with further increase in r. In the case of a turbulent undisturbed boundary layer heat transfer is diminished with increase in r more slowly than in the case of a laminar undisturbed flow. In the presence of an incident shock the bluntness of the leading edge of the flat plate results in a greater decrease in the Stanton number than in the absence of the shock. With increase in the bluntness of the leading edge of the plate the separation zone first sharply lengthens and then decreases in size or remains constant.     

Effect of rotation on the formation of longitudinal vortices in mixed convection flow over a flat plate

This paper presents a numerical study of the effect of rotation on the formation of longitudinal vortices in mixed convection flow over a flat plate. The criterion on the position of marking the onset of longitudinal vortices is defined in this paper. The onset position characterized by the Goertler number G _δ depends on the Grashof number, the rotation number Ro, the Prandtl number Pr and the wave number. The results show that negative rotation stabilizes the boundary layer flow on the surface. On the contrary, positive rotation destabilizes the flow. The numerical data are compared with the experimental results.     

Experimental study of density fluctuations in a hypersonic laminar wake behind a cone

The characteristics of natural fluctuations of density in a laminar near wake behind a sharp cone in a hypersonic flow of nitrogen at zero incidence are studied by the method of electronbeam fluorescence at Mach numberM=21 and unit Reynolds numberRe ₁=6·10⁵ m⁻¹. The distributions of the mean density, integral fluctuations, and spectra of density fluctuations are obtained, the longitudinal and azimuthal phase velocities of perturbations are determined, and the growth rates of perturbations in the wake are found. The results are compared with the measurement data in the shock layer on a flat plate. Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 3, pp. 111–117, May–June, 2000     

Mixed Convection in Water Near the Density Extremum Along a Vertical Plate with Sinusoidal Surface Temperature Variation Embedded in a Porous Medium

A steady laminar boundary layer flowing along a vertical plate immersed in a Darcy–Brinkman porous medium saturated with water at 4°C is studied. The plate temperature varies sinusoidally along the plate between 0 and 8°C where the density of water varies parabolically and is almost symmetrical at about 4°C. Except for the existence of the buoyancy force, it is assumed that either the plate moves upwards or the ambient water moves upwards (moving stream). The results are obtained with the direct numerical solution of the boundary layer equations taking into account the temperature dependence of water thermophysical properties (ρ, μ and c _p). Results are presented for the wall temperature gradient and the wall shear stress along the plate for free convection and mixed convection. Temperature and velocity profiles are also presented.     

Steady mixed convection boundary layer flow over a vertical flat plate in a porous medium filled with water at 4°C: case of variable wall temperature

The problem of steady mixed convection boundary layer flow over a vertical impermeable flat plate in a porous medium saturated with water at 4°C (maximum density) when the temperature of the plate varies as x ^m and the velocity outside boundary layer varies as x ^{2 m} , where x measures the distance from the leading edge of the plate and m is a constant is studied. Both cases of the assisting and the opposing flows are considered. The plate is aligned parallel to a free stream velocity U _∞ oriented in the upward or downward direction, while the ambient temperature is T _∞ = T _m (temperature at maximum density). The mathematical models for this problem are formulated, analyzed and simplified, and further transformed into non-dimensional form using non-dimensional variables. Next, the system of governing partial differential equations is transformed into a system of ordinary differential equations using the similarity variables. The resulting system of ordinary differential equations is solved numerically using a finite-difference method known as the Keller-box scheme. Numerical results for the non-dimensional skin friction or shear stress, wall heat transfer, as well as the temperature profiles are obtained and discussed for different values of the mixed convection parameter λ and the power index m. All the numerical solutions are presented in the form of tables and figures. The results show that solutions are possible for large values of λ and m for the case of assisting flow. Dual solutions occurred for the case of opposing flow with limited admissible values of λ and m. In addition, separation of boundary layers occurred for opposing flow, and separation is delayed for the case of water at 4°C (maximum density) compared to water at normal temperature.     

Exact analytic solutions for free convection boundary layers on a heated vertical plate with lateral mass flux embedded in a saturated porous medium

 The problem of the self-similar boundary flow of a “Darcy-Boussinesq fluid” on a vertical plate with temperature distribution T _w(x) = T _∞+A·x ^λ and lateral mass flux v _w(x) = a·x ^(λ−1)/2, embedded in a saturated porous medium is revisited. For the parameter values λ = 1,−1/3 and −1/2 exact analytic solutions are written down and the characteristics of the corresponding boundary layers are discussed as functions of the suction/ injection parameter in detail. The results are compared with the numerical findings of previous authors. Received on 8 March 1999     

Numerical investigation of natural convection of Al<Subscript>2</Subscript>O<Subscript>3</Subscript> nanofluid in vertical annuli

This paper presents the numerical study of internal free convection of Al₂O₃ water nanofluid in vertical annuli. Vertical walls are maintained at constant temperatures and horizontal walls are adiabatic. Results are validated by experimental data. Effect of nanofluids on natural convection is investigated as a function of geometrical and physical parameters and particle fractions for aspect ratio of 1 ≤ H/L ≤ 5, Grashof number of 10³ ≤ Gr ≤ 10⁵ and concentration of 0 ≤ ϕ ≤ 0.06. More than 330 different numerical cases are investigated to develop a new correlation for the Nusselt number. This correlation is presented as a function of Nusselt number of base fluid and particle fraction which is a linear decreasing function of particle fraction. The developed correlation for annuli is also valid for the natural convection of Al₂O₃ water nanofluid in a square cavity. Furthermore, the effect of the viscosity and conductivity models on the Nusselt number of nanofluids in cylindrical cavities are discussed.     

A correlation for free convection heat transfer from vertical wavy surfaces

Free convection heat transfer along an isothermal vertical wavy surface was studied experimentally and numerically. A Mach-Zehnder Interferometer was used in the experiment to determine the local heat transfer coefficients. Experiments were done for three different amplitude–wavelength ratios of α = 0.05, 0.1, 0.2 and the Rayleigh numbers ranging from Ra _l = 2.9 × 10⁵ to 5.8 × 10⁵. A finite-volume based code was developed to verify the experimental study and obtain the results for all the amplitude–wavelength ratios between α = 0 to 0.2. It is found that the numerical results agree well with the experimental data. Results indicate that the frequency of the local heat transfer rate is the same as that of the wavy surface. The average heat transfer coefficient decreases as the amplitude–wavelength ratio increases and there is a significant difference between the average heat transfer coefficients of the surface with α = 0.2 and those surfaces with α = 0.05 and 0.1. The experimental data are correlated with a single equation which gives the local Nusselt number along the wavy surface as a function of the amplitude–wavelength ratio and the Rayleigh number.     

Effect of local forcing on a turbulent boundary layer

An experimental study is performed to analyze flow structures behind local suction and blowing in a flat-plate turbulent boundary layer. The local forcing is given to the boundary layer flow by means of a sinusoidally oscillating jet issuing from a thin spanwise slot at the wall. The Reynolds number based on the momentum thickness is about Re _θ =1700. The effects of local forcing are scrutinized by altering the forcing frequency (0.011 ≤ f ⁺≤ 0.044). The forcing amplitude is fixed at A ₀=0.4. It is found that a small local forcing reduces the skin friction and the skin friction reduction increases with the forcing frequency. A phase-averaging technique is employed to capture the large-scale vortex evolution. An organized spanwise vortical structure is generated by the local forcing. The cross-sectional area of vortex and the time fraction of vortex are examined by changing the forcing frequency. An investigation of the random fluctuation components reveals that turbulent energy is concentrated near the center of vortical structures. Received: 17 March 2000/Accepted: 3 April 2001     

Nonlinear wave processes in a hypersonic shock layer

Nonlinear disturbance development in a hypersonic flat-plate shock layer (M_∞ = 21, Re _L = 1.44×10⁵) exposed to external-flow slow-mode acoustic perturbations at one or several frequencies is studied on the basis of the numerical solution of the Navier-Stokes equations. The mean flow distortion by disturbances and the nonlinear self-interaction between spectral modes is investigated by varying the initial amplitudes of the acoustic waves introduced. The appearance of combination frequencies, both summarized and subtracted, and their interaction with each other is shown to exist.     

Shock wave standoff distance for a sphere slightly above Mach one

This paper describes a numerical simulation of bow shock formation ahead of a sphere at steady supersonic flow in the Mach number range of 1.025–1.20. Turbulent viscous flow results are presented using the Spalart–Allmaras turbulence model. The purpose of this study is to determine the shock standoff distance for a spherical projectile at slightly supersonic free flight speeds. Results are compared to experimental data, including double exposure holographic interferograms obtained from a 40 mm polycarbonate sphere launched by a light gas gun. The shock standoff distance was determined from the interferograms. The present numerical simulations were found to agree with previously published data, and reached down to M = 1.025—a range where almost no previously published data exists. The computed flow structure and shock wave locations agree well with recently obtained free-flight interferograms.     

Interaction between an Inclined Shock and Boundary and High-Entropy Layers on a Flat Plate

The flow structure and heat exchange in the zone of interference between an inclined shock and the surface of a flat plate are investigated experimentally and theoretically as functions of many parameters, the interference being studied in both the presence and the absence of bluntness of the leading edge. The experiments were carried out at Mach numbers M = 6, 8, and 10 and the Reynolds numbers Re _L , calculated using the plate length L = 120 mm and the free-stream parameters, varied over the range from 0.24 ? 10⁶ to 1.31 ? 10⁶. The bluntness radius of the leading edge of the plate, the intensity of the impinging shock, and its location with respect to the leading edge were varied. The numerical simulation was carried out by solving the complete two-dimensional Navier-Stokes equations and averaged Reynolds equations using the q-ω turbulence model. The laminar boundary layer became turbulent inside the separation zone induced by the shock. It is shown that the plate bluntness significantly reduces the heat exchange intensity in the interference zone, this effect intensifying with increase in the Mach number.     

Axisymmetric shock wave interaction with a cone: a benchmark test

Results of the benchmark test are presented of comparing numerical schemes solving shock wave of M_s = 2.38 in nitrogen and argon interacting with a 43∘ semi-apex angle cone and corresponding experiments. The benchmark test was announced in Shock Waves Vol. 12, No. 4, in which we tried to clarify the effects of viscosity and heat conductivity on shock reflection in conical flows. This paper summarizes results of ten numerical and two experimental applications. State of the art in studies regarding the shock/cone interaction is clarified. PACS 01.50.Kw, 47.15.Pn Communicated by K. Takayama