Numerical modeling of the stabilization of a supersonic flat-plate boundary layer by a porous coating

On the basis of a numerical solution of the two-dimensional Navier-Stokes equations, the stability and the receptivity of a supersonic (M_∞ = 6) boundary layer on a flat plate with a passive porous coating partially absorbing flow disturbances is studied. The results of direct numerical simulation are in good agreement with the data of the linear stability theory. The studies confirm the possibility of effectively stabilizing the second mode of the supersonic boundary layer using porous coatings.     

Experimental study of the effect of a passive porous coating on disturbances in a hypersonic boundary layer. 1. Effect of the porous coating length

The effect of passive porous coatings of different lengths on the second mode of disturbances in a hypersonic boundary layer is considered. The experiments are performed in a flow with a free-stream Mach number M_∞ = 5.8 and five values of the unit Reynolds number around a sharp cone with an apex half-angle equal to 7°, which is aligned at a zero angle of attack. One half of the model surface along its generatrix is covered by a porous material, and the other part is a solid surface. Pressure fluctuations on the model surface are measured. It is found that application of a passive porous coating can either decrease or increase the amplitude of the second mode. The length of the passive porous coating corresponding to the maximum efficiency of its action on flow disturbances and the coating length that increases the amplitude of the second mode are found.     

Numerical simulation of stabilization of the boundary layer on a surface with a porous coating in a supersonic separated flow

Stability of a supersonic (M _∞ = 5.373) boundary layer with local separation in a compression corner with a passive porous coating partly absorbing flow perturbations is considered by solving two-dimensional Navier-Stokes equations numerically. The second mode of disturbances of a supersonic boundary layer is demonstrated to be the most important one behind the boundary-layer reattachment point. The possibility of effective stabilization of these disturbances behind the reattachment point with the use of porous coatings is confirmed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 2, pp. 39–47, March–April, 2007.     

Experimental investigation of the efficiency of gas screens at a burning-out surface with the blowing of a foreign gas

The article gives the results of a systematic experimental investigation of the efficiency of a gas screen at a burning-out graphite surface organized in different manners and with the blowing of a foreign gas into the boundary layer. In the experiments, the Reynolds number was varied within the limits R_Δ ≈ 1.2·10⁵ = 1.6·10⁶, R* ≈ 100–2000, R_Z = 2.3·10²–1.5·10⁴; the enthalpy factor of the nonisothermicity in the initial cross section varied from i_s/i₀ = 1 with the tangential blowing of nitrogen to i_s/i₀ ≈ 34 with the blowing of helium through the heated porous section.     

Numerical simulation of the flow around a porous covering square cylinder in a channel via lattice Boltzmann method

In this paper, a detailed investigation on the flow past a porous covering cylinder is presented through the lattice Boltzmann method. The Brinkman‐Forchheimer‐extended Darcy model is adopted for the entire flow field with the solid, fluid, and porous medium. The effects of several parameters, such as porous layer thickness, Darcy number, porosity, and Reynolds number on flow field are discussed. Compared with the case of a solid cylinder, the present work shows that the porous layer may play an important role on the flow, the lift and drag force exerted on the cylinder. The numerical results indicate that the maximal drag coefficient C_d and maximal amplitude of lift coefficient C_l exist at certain Darcy number which is in the range of 10^?6–10^?2. Copyright © 2010 John Wiley & Sons, Ltd.     

Excitation by sound of Tollmien-Schlichting waves in a supersonic boundary layer

The interaction of sound with a supersonic boundary layer is considered. Because of the dependence of the main flow on the longitudinal coordinate, a sound wave generates unstable oscillations within the boundary layer. Calculations made for Mach number M = 2.0 and dimensionless frequency 2πfv_e/U_e ² = 0.91·10^?4 showed that near the lower branch of the curve of neutral stability a Tollmien—Schlichting wave can be excited with an intensity 2–3 times greater than that of the external acoustic wave.     

Experimental study of the laminar-turbulent transition on a blunt cone

Results of an experimental study of the laminar-turbulent transition in a hypersonic flow around cones with different bluntness radii at a zero angle of attack, free-stream Mach number M _∞ = 6, and unit Reynolds number in the interval Re _∞,1 = 5.79 · 10⁶–5.66 · 10⁷ m^?1 are presented. Flow regimes in which a reverse of the laminar-turbulent transition (decrease in the length of the laminar segment with increasing bluntness radius) are studied. Heat flux distributions over the model surface are obtained with the use of temperature-sensitive paints. Lines of the beginning of the transition in the boundary layer are analyzed by using heat flux fields. The critical Reynolds number Re _∞,R ≈ 1.3 · 10⁵ beginning from which the laminar-turbulent transition substantially depends on uncontrolled disturbances, such as the model tip roughness, is found. In supercritical regimes, the line of the transition beginning is shifted in most cases toward the model tip (reverse of the transition). The results obtained are compared with available experimental data.     

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     

Effect of sound-absorbing materials on intensity of disturbances in the shock layer on a flat plate aligned at an angle of attack

Results of a numerical and experimental study of characteristics of disturbances in a hypersonic shock layer on a flat plate covered by a sound-absorbing coating and aligned at an angle of attack are presented. Experiments and computations are performed for the free-stream Mach number M _∞ = 21 and Reynolds number Re _L = 6 · 10⁴. A possibility of suppressing pressure fluctuations in the shock layer at frequencies of 20–40 kHz with the use of tubular and porous materials incorporated into the plate surface is demonstrated. Results of numerical simulations are found to be in good agreement with experimental data.     

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.     

Experimental study of evolution of disturbances in a supersonic boundary layer on a swept-wing model under controlled conditions

Experimental data on stability of a three-dimensional supersonic boundary layer on a swept wing are presented. Evolution of artificial wave trains was studied. The experiments were conducted for Mach numberM=2.0 and unit Reynolds numberRe ₁=6.6·10⁶m⁻¹ on a swept-wing model with a lenticular profile and a40° sweep angle of the leading edge at zero incidence. Excitation of high-frequency disturbances caused by secondary-flow instability at a high initial amplitude was observed. It is shown that the evolution of disturbances at frequencies of10, 20, and30 kHz is similar to the development of travelling waves for the case of subsonic velocities. Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 1, pp. 50–56, January–February, 2000.     

Dynamics of boundary layer formation in a volcano channel in a cavitating high-viscosity magma flow

Based on the full mathematical model of a viscous magma melt flow ascending in the gravity field behind a decompression wave front, an unsteady two-dimensional axisymmetric problem of the melt state dynamics at the initial stage of an explosive volcanic eruption and specific features of the flow in the vicinity of the channel wall for the cases of stationary and dynamically increasing viscosity are studied. The evolution of the boundary layer is numerically analyzed for a constant melt viscosity equal to μ = 10 ³ , 10 ⁵ , and 10 ⁷ Pa · sec. It is demonstrated that a boundary layer is formed on the wall of the channel with a radius of 100 m as the melt viscosity is changed in the range of 10 ³ –10 ⁵ Pa · sec, and the boundary layer thickness increases from 2 to 15 m. As the magma viscosity increases to 10 ⁷ Pa · sec, the boundary layer chokes the major part of the channel, thus, locking the flow in the vicinity of the axis of symmetry of the channel almost over the entire channel length. Substantial changes in the flow structure caused by dynamically increasing viscosity are demonstrated by an example of the melt in the channel with a radius of 10 m. By the time t = 1.1 sec, the boundary layer thickness in the channel cross section at a height of approximately 1000 m reaches almost 8 m, the boundary layer acquires the shape similar to a “diaphragm,” penetrates inward the channel by 200 m (with the mass velocity ranging from 0 to 15 m/sec), and locks the flow in a zone with a radius of approximately 2 m around the axis of symmetry of the channel.     

Stability of a nonuniformly heated fluid in a porous horizontal layer

We investigate the stability of a nonuniformly heated fluid in the gravitational field in a plane horizontal porous layer through which vertical forced motion is effected. A similar system was studied in [1, 2]. In the present paper, the nonuniformity of the permeability of the porous layer with respect to the depth and the dependence of the viscosity of the saturating fluid on the temperature are taken into account in addition. As a result of the application of the linear stability theory, an eigenvalue problem arises, which is solved numerically. A family of curves representing the dependence of the critical modified Rayleigh number Ra _k ^⋆ on the injection parameter (the Péclet number Pe) for different degrees of inhomogeneity of the permeability and the viscosity is obtained. It is found that although Pe=0 corresponds to Ra _k ^⋆ for uniform permeability and viscosity and the stability increases monotonically as Pe increases, the presence of nonuniformity of the permeability or the viscosity leads to the appearance of a stability minimum in the region Pe≈1, while under the simultaneous influence of these two factors, the minimum is shifted into the region Pe≈2. The results of the paper can be used, for example, in the investigation of heat transfer in the case of forced fluid motion in the fissures of a permeable rock mass, when, in the case of pumping through a horizontal fissure, the fluid penetrates vertically across its permeable walls into the stratum. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 3–7, November–December, 1986.     

Natural Convection in a Cavity Filled with Porous Layers on the Top and Bottom Walls

Natural convection in a partially filled porous square cavity is numerically investigated using SIMPLEC method. The Brinkman-Forchheimer extended model was used to govern the flow in the porous medium region. At the porous-fluid interface, the flow boundary condition imposed is a shear stress jump, which includes both the viscous and inertial effects, together with a continuity of normal stress. The thermal boundary condition is continuity of temperature and heat flux. The results are presented with flow configurations and isotherms, local and average Nusselt number along the cold wall for different Darcy numbers from 10⁻¹ to 10⁻⁶, porosity values from 0.2 to 0.8, Rayleigh numbers from 10³ to 10⁷, and the ratio of porous layer thickness to cavity height from 0 to 0.50. The flow pattern inside the cavity is affected with these parameters and hence the local and global heat transfer. A modified Darcy–Rayleigh number is proposed for the heat convection intensity in porous/fluid filled domains. When its value is less than unit, global heat transfer keeps unchanged. The interfacial stress jump coefficients β ₁ and β ₂ were varied from  −1 to +1, and their effects on the local and average Nusselt numbers, velocity and temperature profiles in the mid-width of the cavity are investigated.     

Flow stability in a channel with porous walls

We study the stability of the flow which forms in a plane channel with influx of an incompressible viscous fluid through its porous parallel walls. Under certain assumptions the study of the stability reduces to the solution of modified Orr-Sommerfeld equation accounting for the transverse component of the main-flow velocity. As a result of numerical integration of this equation we find the dependence of the local critical Reynolds number on the blowing Reynolds number R₀, which may be defined by two factors: the variation of the longitudinal velocity profile with R₀ and the presence of the transverse velocity component. A qualitative comparison is made of the computational results with experimental data on transition from laminar to turbulent flow regimes in channels with porous walls, which confirms that it is necessary to take into account the effect of the transverse component of the main-flow velocity on the main-flow stability in the problem in question.Flows in channels with porous walls are of interest for hydrodynamic stability theory in view of the fact that they can be described by the exact solutions of the Navier-Stokes equations by analogy with the known Poiseuille and Couette flows. However, in contrast with the latter, the flows in channels with porous walls (studies in [1], for example) will be nonparallel.The theory of hydrodynamic stability of parallel flows has frequently been applied to nonparallel flows (in the boundary layer, for example). In so doing the nonparallel nature of the flow has been taken into account only by varying the longitudinal velocity component profiles. A study was made in [2, 3] of the effect of the transverse component of the main flow on its stability. In the case of the boundary layer in a compressible gas, a considerable influence of the transverse velocity component on the critical Reynolds number was found in [2] and confirmed experimentally. A strong influence of the transverse velocity component on the instability region was also found in [3] in a study of the flow stability in a boundary layer with suction for an incompressible fluid.     

Linear stability of a fluid channel with a porous layer in the center简

We perform a Poiseuille flow in a channel linear stability analysis of a inserted with one porous layer in the centre, and focus mainly on the effect of porous filling ratio. The spectral collocation technique is adopted to solve the coupled linear stability problem. We investigate the effect of permeability, σ, with fixed porous filling ratio ψ = 1/3 and then the effect of change in porous filling ratio. As shown in the paper, with increasing σ, almost each eigenvalue on the upper left branch has two subbranches at ψ = 1/3. The channel flow with one porous layer inserted at its middle （ψ = 1/3） is more stable than the structure of two porous layers at upper and bottom walls with the same parameters. By decreasing the filling ratio ψ, the modes on the upper left branch are almost in pairs and move in opposite directions, especially one of the two unstable modes moves back to a stable mode, while the other becomes more instable. It is concluded that there are at most two unstable modes with decreasing filling ratio ψ. By analyzing the relation between ψ and the maximum imaginary part of the streamwise phase speed, Cimax, we find that increasing Re has a destabilizing effect and the effect is more obvious for small Re, where ψ a remarkable drop in Cimax can be observed. The most unstable mode is more sensitive at small filling ratio ψ, and decreasing ψ can not always increase the linear stability. There is a maximum value of Cimax which appears at a small porous filling ratio when Re is larger than 2 000. And the value of filling ratio 0 corresponding to the maximum value of Cimax in the most unstable state is increased with in- creasing Re. There is a critical value of porous filling ratio （= 0.24） for Re = 500; the structure will become stable as ψ grows to surpass the threshold of 0.24; When porous filling ratio ψ increases from 0.4 to 0.6, there is hardly any changes in the values of Cimax. We have also observed that the critical Reynolds number is especially sensitive for small ψ where the fastest drop is observed, and there may be a wide range in which the porous filling ratio has less effect on the stability （ψ ranges from 0.2 to 0.6 at σ = 0.002）. At larger permeability, σ, the critical Reynolds number tends to converge no matter what the value of porous filling ratio is.     

A transition phenomenon of ice formation in water flow between two horizontal parallel plates

Experiments have been performed to investigate the icetransition profiles and heat-transfer characteristics for water flows between two horizontal parallel plates. The experiments are carried out under the condition that upper plate is cooled at uniform temperature kept less than freezing temperature of water, while the lower plate is heated at uniform temperature kept higher than the temperature of water flow. The temperatures of the upper and lower plates range from ?8 to ?14°C and from 10 to 60 °C, respectively, with inlet-water temperature varied from 1.5 to 4.5 °C. The cooling and heating temperature ratios, θ_c and θ_h, are ranging from 1.78 to 9.33 and from 1.22 to 39, respectively. By using three kinds of heightH of 16, 30 and 40 mm between the horizontal parallel plates, the Reynolds and Grashof numbers are varied from 3.2 × 10² to 1.5 × 10⁴ and from 3.4 × 10³ to 8.97 × 10⁶, respectively. As a result of this investigation two ice-transition modes are observed. The first ice-transition mode is due to an interruption of upper and lower thermal boundary layers, while the second mode is due to an instability of laminar boundary layer formed on water-ice interface. In order to determine the kind of ice-transition mode, criterion correlation formulas including the Reynolds numberRe _H , Grashof numberGr _H , and heating temperature ratio θ_h are determined and may be written as follows: For thermal icetransition mode (th.I.T.M.)Re _H /(Gr _H ·θ _h )^0.23<1.6×10^?3 and for hydrodynamical ice-transition mode (hy.I.T.M.)Re _H /(Gr _H ·θ _h )^0.23>2.3×10^?3 By introducing the freezing parameterB _f , correlation equations for local and mean Nusselt numbers along the water-ice interface at steady-state condition are determined. From the current experimental results it is found that the local Nusselt number may be described as the following equation:Nu _x =0.835 Re _H ^0.278 · B _f ^0.834 ·x/H)^?0.139     

The onset of buoyancy-driven convection in a ferromagnetic fluid saturated porous medium

The onset of buoyancy-driven convection in an initially quiescent ferrofluid saturated horizontal porous layer in the presence of a uniform vertical magnetic field is investigated. The Brinkman-Lapwood extended Darcy equation with fluid viscosity different from effective viscosity is used to describe the flow in the porous medium. The lower boundary of the porous layer is assumed to be rigid-paramagnetic, while the upper paramagnetic boundary is considered to be either rigid or stress-free. The thermal conditions include fixed heat flux at the lower boundary, and a general convective–radiative exchange at the upper boundary, which encompasses fixed temperature and fixed heat flux as particular cases. The resulting eigenvalue problem is solved numerically using the Galerkin technique. It is found that increase in the Biot number Bi, porous parameter σ, viscosity ratio Λ, magnetic susceptibility χ, and decrease in the magnetic number M ₁ and non-linearity of magnetization M ₃ is to delay the onset of ferroconvection in a porous medium. Further, increase in M ₁, M ₃, and decrease in χ, Λ, σ and Bi is to decrease the size of convection cells.     

Effect of the cone nose bluntness and the ultrasonically absorptive coating on transition in a hypersonic boundary layer

The effect of an ultrasonically-absorptive coating on laminar-turbulent transition on cones with different nose bluntnesses is experimentally investigated. The experiments were performed with a cone with the semi-vertex angle of 7° set at zero incidence in the Mach 8 flow for three Reynolds numbers. A material with a chaotic micropore structure was used as the ultrasonically-absorptive coating. One side of the model, along its generator, was coated with the porous material, while the second represented a rigid surface. The laminar-turbulent transition location was determined from the results of heat flux distribution measurements. The heat flux fluctuations were also measured on the model surface. It was found that the laminar region length increased with an increase in the bluntness radius. The ultrasonically-absorptive coating with a chaotic microstructure effectively stabilizes the boundary layer for all bluntness radii considered, increasing the laminar region length by 30 to 85%.     

Velocity and temperature distributions on a semi-infinite flat plate embedded in a saturated porous medium

In this paper the velocity and temperature distributions on a semi-infinite flat plate embedded in a saturated porous medium are obtained for the governing equations (Kaviany [7]) following the technique adopted by Chandrashekara [2] which are concerned with the interesting situations of the existence of transverse, velocity and thermal boundary layers. Here the pressure gradient is just balanced by the first and second order solid matrix resistances for small permeability and observed that by increasing of the flow resistance the asymptotic value for the heat transfer rate increases. Further we concluded that the transverse boundary layers are thicker than that of axial boundary layers. Hence we evaluated the expressions for the boundary layer thickness, the shear stress at the semi-infinite plate and _T (the ratio of the thicknesses of the thermal boundary layer and momentum boundary layer). The variations of these quantities for different values of the porous parameterB and the flow resistanceF have been discussed in detail with the help of tables. The curves for velocity and temperature distributions have been plotted for different values ofB andF.Lastly we have evaluated the heat fluxq(x) and found that it depends entirely upon the Reynolds numberRe, Prandtl numberPr,B andF.