Laminar-turbulent transition in the boundary layer on cones in a hypersonic flow at high Reynolds numbers per meter

The laminar-turbulent transition is experimentally studied in boundary-layer flows on cones with a rectangular axisymmetric step in the base part of the cone and without the step. The experiments are performed in an A-1 two-step piston-driven gas-dynamic facility with adiabatic compression of the working gas with Mach numbers at the nozzle exit M _∞ = 12–14 and pressures in the settling chamber P₀ = 60–600 MPa. These values of parameters allow obtaining Reynolds numbers per meter near the cone surface equal to Re _1e = (53–200) · 10⁶ m ⁻¹. The transition occurs at Reynolds numbers Re _tr = (2.3–5.7) · 10⁶. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 76–83, May–June, 2007.     

Numerical study of transient laminar natural convection heat transfer over a sphere subjected to a constant heat flux

Transient laminar natural convection over a sphere which is subjected to a constant heat flux has been studied numerically for high Grashof numbers (10⁵ ≤ Gr ≤ 10⁹) and a wide range of Prandtl numbers (Pr = 0.02, 0.7, 7, and 100). A plume with a mushroom-shaped cap forms above the sphere and drifts upward continuously with time. The size and the level of temperature of the transient cap and plume stem decrease with increasing Gr and Pr. Flow separation and an associated vortex may appear in the wake of the sphere depending on the magnitude of Gr and Pr. A recirculation vortex which appears and grows until “steady state” is attained was found only for the very high Grashof numbers (10⁵ ≤ Gr ≤ 10⁹) and the lowest Prandtl number considered (Pr = 0.02). The appearance and subsequent disappearance of a vortex was observed for Gr = 10⁹ and Pr = 0.7. Over the lower hemisphere, the thickness of both the hydrodynamic (δ_H) and the thermal (δ_T) boundary layers remain nearly constant and the sphere surface is nearly isothermal. The surface temperature presents a local maximum in the wake of the sphere whenever a vortex is established in the wake of the sphere. The surface pressure recovery in the wake of the sphere increases with decreasing Pr and with increasing Gr. For very small Pr, unlike forced convection, the ratio δ_T/δ_H remains close to unity. The results are in good agreement with experimental data and in excellent agreement with numerical results available in the literature. A correlation has also been presented for the overall Nusselt number as a function of Gr and Pr.     

Influence of Mach and Reynolds numbers on heat transfer to the leeward surface of a half-cone at hypersonic speed

Results are given of an investigation of heat transfer on the flat surface of a blunted half-cone, washed at zero angle of attack by a helium flow at high Mach number (up to 23.5). A correlation is given for the experimental data obtained over a wide range of Mach numbers (M = 3–23.5) and Reynolds numbers (Re_a = 10⁴–3.5·⁵, wherea is the nose radius).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 105–109, September–October, 1976.     

Control of Flow and Heat Transfer in a Porous Enclosure due to an Adiabatic Thin fin on the Hot Wall

Natural convection flow in a differentially heated square enclosure filled with porous matrix with a solid adiabatic thin fin attached at the hot left wall is studied numerically. The Brinkman–Forchheimer-extended Darcy model is used to solve the momentum equations, in the porous medium. The numerical investigation is done through streamlines, isotherms, and heat transfer rates. A parametric study is carried out using the following parameters: Darcy number (Da) from 10⁻⁴ to 10⁻², dimensionless thin fin lengths (L _p) 0.3, 0.5, and 0.7, dimensionless positions (S _p) 0.25, 0.5, and 0.75 with Prandtl numbers (Pr) 0.7 and 100 for Ra = 10⁶. For Da = 10⁻³ and Pr = 0.7, it is observed that there is a counter clock-wise secondary flow formation around the tip of the fin for S _p = 0.5 for all lengths of L _p. Moreover when Da = 10⁻² the secondary circulation behavior has been observed for S _p = 0.25 and 0.75 and there is another circulation between the top wall and the fin that is separated from the primary circulation. However, these secondary circulations features are not observed for Pr = 100. It is also found that the average Nusselt number decreases as the length of the fin increases for all locations. However, the rate of decrease of average Nusselt number becomes slower as the location of fin moves from the bottom wall to the top wall. The overall heat transfer rate can be controlled with a suitable selection of the fin location and length.     

L2Roe: a low dissipation version of Roe's approximate Riemann solver for low Mach numbers

A modification of the Roe scheme called L²Roe for low dissipation low Mach Roe is presented. It reduces the dissipation of kinetic energy at the highest resolved wave numbers in a low Mach number test case of decaying isotropic turbulence. This is achieved by scaling the jumps in all discrete velocity components within the numerical flux function. An asymptotic analysis is used to show the correct pressure scaling at low Mach numbers and to identify the reduced numerical dissipation in that regime. Furthermore, the analysis allows a comparison with two other schemes that employ different scaling of discrete velocity jumps, namely, LMRoe and a method of Thornber et al. To this end, we present for the first time an asymptotic analysis of the last method. Numerical tests on cases ranging from low Mach number (M_∞=0.001) to hypersonic (M_∞=5) viscous flows are used to illustrate the differences between the methods and to show the correct behavior of L²Roe. No conflict is observed between the reduced numerical dissipation and the accuracy or stability of the scheme in any of the investigated test cases. Copyright © 2015 John Wiley & Sons, Ltd.     

Direct numerical simulation of turbulent flow and heat transfer in a square duct with natural convection

In this paper, a direct numerical simulation of a fully developed turbulent flow and heat transfer are studied in a square duct with an imposed temperature difference between the vertical walls and the perfectly insulated horizontal walls. The natural convection is considered on the cross section in the duct. The numerical scheme employs a time-splitting method to integrate the three dimensional incompressible Navier-Stokes equation. The unsteady flow field was simulated at a Reynolds number of 400 based on the Mean friction velocity and the hydraulic diameter (Re _m = 6200), while the Prandtl number (Pr) is assumed 0.71. Four different Grashof numbers (Gr = 10⁴, 10⁵, 10⁶ and 10⁷) are considered. The results show that the secondary flow and turbulent characteristics are not affected obviously at lower Grashof number (Gr ≤ 10⁵) cases, while for the higher Grashof number cases, natural convection has an important effect, but the mean flow and mean temperature at the cross section are also affected strongly by Reynolds stresses. Compared with the laminar heat transfer at the same Grashof number, the intensity of the combined heat transfer is somewhat decreased.     

Simulation of turbulent flows around a circular cylinder using nonlinear eddy-viscosity modelling: steady and oscillatory ambient flows

Steady incident flow past a circular cylinder for sub- to supercritical Reynolds number has been simulated as an unsteady Reynolds-averaged Navier–Stokes (RANS) equation problem using nonlinear eddy-viscosity modelling assuming two-dimensional flow. The model of Craft et al. (Int. J. Heat Fluid Flow 17 (1996) 108), with adjustment of the coefficients of the ‘cubic’ terms, predicts the drag crisis at a Reynolds number of about 2×10⁵ due to the onset of turbulence upstream of separation and associated changes in Strouhal number and separation positions. Slightly above this value, at critical Reynolds numbers, drag is overestimated because attached separation bubbles are not simulated. These do not occur at supercritical Reynolds numbers and drag coefficient, Strouhal number and separation positions are in approximate agreement with experimental measurements (which show considerable scatter). Fluctuating lift predictions are similar to sectional values measured experimentally for subcritical Reynolds numbers but corresponding measurements have not been made at supercritical Reynolds numbers. For oscillatory ambient flow, in-line forces, as defined by drag and inertia coefficients, have been compared with the experimental values of Sarpkaya (J. Fluid Mech. 165 (1986) 61) for values of the frequency parameter, β=D²/νT, equal to 1035 and 11240 and Keulegan–Carpenter numbers, KC=U₀T/D, between 0.2 and 15 (D is cylinder diameter, ν is kinematic viscosity, T is oscillation period, and U₀ is the amplitude of oscillating velocity). Variations with KC are qualitatively reproduced and magnitudes show best agreement when there is separation with a large-scale wake, for which the turbulence model is intended. Lift coefficients, frequency and transverse vortex shedding patterns for β=1035 are consistent with available experimental information for β≈250−500. For β=11240, it is predicted that separation is delayed due to more prominent turbulence effects, reducing drag and lift coefficients and causing the wake to be more in line with the flow direction than transverse to it. While these oscillatory flows are highly complex, attached separation bubbles are unlikely and the flows probably two dimensional.     

Unsteady free convection on a vertical cylinder with variable heat and mass flux

The unsteady natural convection boundary layer flow over a semi-infinite vertical cylinder is considered with combined buoyancy force effects, for the situation in which the surface temperature T ^′ _w(x) and C ^′ _w(x) are subjected to the power-law surface heat and mass flux as K(T ^′/r) = −ax ⁿ and D(C ^′/r) = −bx ^m . The governing equations are solved by an implicit finite difference scheme of Crank-Nicolson method. Numerical results are obtained for different values of Prandtl number, Schmidt number ‘n’ and ‘m’. The velocity, temperature and concentration profiles, local and average skin-friction, Nusselt and Sherwood numbers are shown graphically. The local Nusselt and Sherwood number of the present study are compared with the available result and a good agreement is found to exist. Received on 7 July 1998     

Experimental investigation of mixed convection from an array of discrete heat sources at the bottom of a horizontal channel

Mixed convection heat transfer from an array of discrete heat sources inside a rectangular channel has been investigated experimentally under various operating conditions for air. The lower surface of the channel was equipped with 8 × 4 flush-mounted heat sources subjected to uniform heat flux, sidewalls and the upper wall are insulated and adiabatic. The experimental parametric study was made for an aspect ratio of AR = 10, Reynolds numbers 241 Re_Dh 980, and modified Grashof numbers Gr^* = 9.53 × 10⁵ to 1.53 × 10⁷ . From the experimental measurements, surface temperature distributions of the discrete heat sources were obtained and effects of Reynolds and Grashof numbers on these temperatures were investigated. Furthermore, Nusselt number distributions were calculated for different Reynolds and Grashof numbers, with emphasis on changes obtained for different discrete heat source locations. From these results, the buoyancy affected secondary flow and the onset of instability have been discussed. Results show that surface temperatures increase with increasing Grashof number and decrease with increasing Reynolds number. However, with the increase in the buoyancy affected secondary flow and the onset of instability, temperatures level off and even drop as a result of heat transfer enhancement. This outcome can also be observed from the variation of the row-averaged Nusselt number showing an increase towards the exit, especially for low Reynolds numbers.     

Experimental investigations of the stability limit of the helical flow of pseudoplastic liquids

The stability of the laminar helical flow of pseudoplastic liquids has been investigated with an indirect method consisting in the measurement of the rate of mass transfer at the surface of the inner rotating cylinder. The experiments have been carried out for different values of the geometric parameter = R ₁/R ₂ (the radius ratio) in the range of small values of the Reynolds number,Re < 200. Water solutions of CMC and MC have been used as pseudoplastic liquids obeying the power law model. The results have been correlated with the Taylor and Reynolds numbers defined with the aid of the mean viscosity value. The stability limit of the Couette flow is described by a functional dependence of the modified critical Taylor number (including geometric factor) on the flow indexn. This dependence, general for pseudoplastic liquids obeying the power law model, is close to the previous theoretical predictions and displays destabilizing influence of pseudoplasticity on the rotational motion. Beyond the initial range of the Reynolds numbers values (Re>20), the stability of the helical flow is not affected considerably by the pseudoplastic properties of liquids. In the range of the monotonic stabilization of the helical flow the stability limit is described by a general dependence of the modified Taylor number on the Reynolds number. The dependence is general for pseudoplastic as well as Newtonian liquids.Nomenclature C _i concentration of reaction ions, kmol/m³ - d = R ₂ –R ₁ gap width, m - F _M () Meksyn's geometric factor (Eq. (1)) - F ₀ Faraday constant, C/kmol - i _l density of limit current, A/m³ - k _c mass transfer coefficient, m/s - n flow index - R ₁,R ₂ inner, outer radius of the gap, m - Re = V _m ·2d·/µ _m Reynolds number - Ta _c = _c ·d^3/2·R ₁ ^1/2 ·/µ _m Taylor number - Z _i number of electrons involved in electrochemical reaction - = R ₁/R ₂ radius ratio - µ apparent viscosity (local), Ns/m² - µ _m mean apparent viscosity value (Eq. (3)), Ns/m² - µ _i apparent viscosity value at a surface of the inner cylinder, Ns/m² - density, kg/m³ - _c angular velocity of the inner cylinder (critical value), 1/s     

Multilevel bridged crack model for high-performance concretes

A multilevel bridged crack model is proposed. It reproduces the constitutive flexural response of reinforced concrete members with fibers. Considered are two different reinforcements: the longitudinal bars (primary reinforcement) and the fibers (secondary reinforcement) distributed in the brittle cementitious matrix. The bridging actions exerted by the reinforcements onto the crack faces are assumed to be rigid-perfectly plastic as the primary constituents. Cohesive softening applies to the fibers.From dimensional analysis, the constitutive flexural response is found to depend on three dimensionless parameters. The first , controls the extension of the process zone. The remaining two parameters, referred to as brittleness numbers N_P⁽¹⁾ and N_P⁽²⁾, are related to the reinforcement phases. Specimen size scale is basic to the global structural behaviour. It can range from ductile to brittle as characterized by the two brittleness numbers. They depend on the reinforcement phase of matrix toughness, reinforcement yielding or slippage limit, reinforcement volume fraction and global structural size.     

Laminar stability analysis of condensate film flow

The linear stability theory is used to study stability characteristics of laminar gravity-induced condensate film flow down an arbitrarily inclined wall. The coupled equations describing the velocity and temperature disturbances are solved numerically. The results show that laminar condensate films are unstable in all practical situations. Several stabilizing effects are acting on the film flow; these are: the angle of inclination, the surface tension at large wave numbers, the condensation rate at small Reynolds numbers, and to a certain extent the Prandtl number. For a vertical plate, the expected wavelengths of the disturbances are presented as functions of the Reynolds numbers of the condensate flow.

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Zusammenfassung Mit Hilfe der linearen StabilitÄtstheorie werden die StabilitÄtseigenschaften laminarer Kondensatfilme an ebenen WÄnden untersucht. Die Gleichungssysteme, die Temperatur- und Geschwindigkeitsstörungen beschreiben, werden numerisch gelöst. Es zeigt sich, da\ Kondensatfilme in jedem praktischen Fall ein unstabiles Verhalten aufweisen. Der stabilisierende Einflu\ von OberflÄchenspannung, Schwerkraft und Stoffübertragung durch Kondensation werden diskutiert. Für eine senkrechte Wand werden die zu erwartenden WellenlÄngen der Störungen als Funktion der Reynoldszahlen des Kondensatfilms angegeben.

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Abrreviations

Nomenclature C^*=C _r ^* + iC _i ^* dimensional complex wave velocity - C=C^*/u₀ dimensionless wave velocity - c_p specific heat at constant pressure - g gravitational acceleration - h_n defined by Eq. (16) - h_fg latent heat - k thermal conductivity - Pe=PrRe Peclet number - Pr Prandtl number - P_y defined by Eq. (15) - q iaPe - Re=u₀ Reynolds number - S temperature disturbance amplitude - t^* dimensional time - t=t^* u₀/ dimensionless time - T dimensional temperature - T_s saturation temperature - T_w wall temperature - T =T_s–T_w temperature drop across liquid film - u^*, v^* dimensional velocity component - v=v^*/u₀ dimensionless velocity components - u₀ dimensional surface velocity of undisturbed film flow - x^*, y^* dimensional coordinates - x=x^*/ dimensionless coordmates - Y_n functional vector defined by Eq. (20) Greek Symbols dimensionless wave number - roots of Eq. (20) - n defined by Eq. (21) - local thickness of undisturbed condensate film - ^* wavelength, dimensional - wavelength, dimensionless - temperature variable - kinematic viscosity of liquid - liquid density - _g vapor density - surface tension - stream function disturbance amplitude - stream function - angle of inclination     

Numerical Simulation of Thermal and Concentration Natural Convection in a Porous Cavity in the Presence of an Opposing Flow

Two-dimensional steady-state thermal concentration convection in a rectangular porous cavity is simulated numerically. The temperature and concentration gradients are horizontal and the buoyancy forces act either in the same or in opposite directions. The flow through the porous medium is described by the Darcy-Brinkman or Forchheimer equations. The SIMPLER numerical algorithm based on the finite volume approach is used for solving the problem in the velocity-pressure variables.Numerous series of calculations were carried out over the range Ra _t =3·10⁶ and 3·10⁷, 10^-6 < Da < 1, 1 < N < 20, Le=10 and 100, where Ra, Da, Le, and N are the Rayleigh, Darcy, and Lewis numbers and the buoyancy ratio, respectively. It is shown that the main effect of the presence of the porous medium is to reduce the heat and mass transfer and attenuate the flow field with decrease in permeability. For a certain combination of the Ra, Le, and N numbers the flow has a multicellular structure. The mean Nusselt and Sherwood numbers are presented as functions of the governing parameters.     

Onset of Buoyancy-driven Instability in Porous Media Melted from Below

When a nonhomogeneous solid is melting from below, convection may be induced in a thermally–unstable melt layer. In this study, the onset of buoyancy-driven convection during time-dependent melting is investigated by using similarly transformed disturbance equations. The critical Darcy–Rayleigh numbers based on the melt-layer thickness, Ra _H,c, are found numerically for various conditions. For small superheats, the present predictions show that Ra _H,c is located between 27.1 and 4π ² and it approaches the well-known results of the original Horton–Rogers–Lapwood problem. However, for high superheats, it is dependent on the phase change rate λ and the relation of Ra _H,c λ = 25.89 is shown.     

Experimental study of the shock–bubble interaction with reshock

The interaction of a planar shock wave with a spherical density inhomogeneity is studied experimentally under reshock conditions. Reshock occurs when the incident shock wave, which has already accelerated the spherical bubble, reflects off the tube end wall and reaccelerates the inhomogeneity for a second time. These experiments are performed at the Wisconsin Shock Tube Laboratory, in a 9m-long vertical shock tube with a large square cross section (25.4×25.4 cm²). The bubble is prepared on a pneumatically retracted injector and released into a state of free fall. Planar diagnostic methods are used to study the bubble morphology after reshock. Data are presented for experiments involving two Atwood numbers (A = 0.17 and 0.68) and three Mach numbers (1.35 < M < 2.33). For the low Atwood number case, a secondary vortex ring appears immediately after reshock which is not observed for the larger Atwood number. The post-reshock vortex velocity is shown to be proportional to the incident Mach number, M, the initial Atwood number, A, and the incident shock wave speed, W _i.     

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     

Investigation of Dynamic Stability Regimes for Axisymmetric Bodies in Supersonic Flow

The problem of supersonic flow past a slender blunt cone with allowance for the reverse boundary-layer effect on the outer flow is solved with the aim of studying the influence of the boundary layer on the damping coefficient of axisymmetric body oscillations. It is assumed that the body executes plane angular, both low-amplitude and low-velocity, oscillations about a center of rotation. A modified version of the method [1] is applied for calculating the time-dependent flow past a body with the viscosity effect taken into account. The high accuracy of the flow parameter determination provided by this technique is confirmed by wind- tunnel experiments on a large-scale cone model (L1 m) at Mach numbers M=4 and 6. The agreement between the calculated and measured data forms the basis for the numerical investigation of the blunt-cone damping coefficient over a wide range of freestream Mach (M=4–20) and Reynolds (Re _L =10⁶–10⁸) numbers. At moderate freestream Mach numbers (M=4 and 6) an appreciable Re _L effect on the damping coefficient was not detected. However, on the hypersonic range this effect manifests itself more strongly, especially when there is gas injection into the boundary layer from the vehicle surface.     

Numerical study of the effects occurring near a circular cylinder in stratified fluid flows with short buoyancy periods

Flow around a two-dimensional circular cylinder of a stratified fluid with periods buoyancy T_b = 25.2 and 6.28 sec is studied numerically over a wide range of Reynolds and Froude numbers. It is found that in the presence of a perturbation ahead of a cylinder which moves downstream with increasing Reynolds number, the salinity isolines have the shape of a semi-circular comb with sharp teeth. The shape change of the attached waves and the occurrence of fluid layers of different densities in the cylinder wake are studied. In flows with a buoyancy period T_b = 6.28 sec at Reynolds numbers Re < 60, stagnant zones are found in the cylinder wake, and at Re > 60, these zones are absent.