Nonsimilarity solutions for non-Darcy mixed convection from horizontal surfaces in a porous medium

Non-Darcy mixed convection in a porous medium from horizontal surfaces with variable surface heat flux of the power-law distribution is analyzed. The entire mixed convection regime is divided into two regions. The first region covers the forced convection dominated regime where the dimensionless parameter ζ _f =Ra^* _x /Pe² _x is found to characterize the effect of buoyancy forces on the forced convection with K ^′ U _∞/ν characterizing the effect of inertia resistance. The second region covers the natural convection dominated regime where the dimensionless parameter ζ _n =Pe _x /Ra^*1/2 _x is found to characterize the effect of the forced flow on the natural convection, with (K ^′ U _∞/ν)Ra^*1/2 _x /Pe _x characterizing the effect of inertia resistance. To obtain the solution that covers the entire mixed convection regime the solution of the first regime is carried out for ζ _f =0, the pure forced convection limit, to ζ _f =1 and the solution of the second is carried out for ζ _n =0, the pure natural convection limit, to ζ _n =1. The two solutions meet and match at ζ _f =ζ _n =1, and R ^* _h =G ^* _h . Also a non-Darcy model was used to analyze mixed convection in a porous medium from horizontal surfaces with variable wall temperature of the power-law form. The entire mixed convection regime is divided into two regions. The first region covers the forced convection dominated regime where the dimensionless parameter ξ _f =Ra _x /Pe _x ^3/2 is found to measure the buoyancy effects on mixed convection with Da _x Pe _x /ɛ as the wall effects. The second region covers the natural convection dominated region where ξ _n =Pe _x /Ra _x ^2/3 is found to measure the force effects on mixed convection with Da _x Ra _x ^2/3/ɛ as the wall effects. Numerical results for different inertia, wall, variable surface heat flux and variable wall temperature exponents are presented. Received on 8 July 1996     

Non-Darcy mixed convection from a vertical plate in saturated porous media-variable surface heat flux

Nonsimilarity solutions for non-Darcy mixed convection from a vertical impermeable surface embedded in a saturated porous medium are presented for variable surface heat flux (VHF) of the power-law form. The entire mixed convection region is divided into two regimes. One region covers the forced convection dominated regime and the other one covers the natural convection dominated regime. The governing equations are first transformed into a dimensionless form by the nonsimilar transformation and then solved by a finite-difference scheme. Computations are based on Keller Box method and a tolerance of iteration of 10⁻⁵ as a criterion for convergence. Three physical aspects are introduced. One measures the strength of mixed convection where the dimensionless parameter Ra^* _x /Pe^3/2 _x characterizes the effect of buoyancy forces on the forced convection; while the parameter Pe _x /Ra^*2/3 _x characterizes the effect of forced flow on the natural convection. The second aspect represents the effect of the inertial resistance where the parameter K′U _∞/ν is found to characterize the effect of inertial force in the forced convection dominated regime, while the parameter (K′U _∞/ν)(Ra^*2/3 _x /Pe _x ) characterizes the effect of inertial force in the natural convection dominated regime. The third aspect is the effect of the heating condition at the wall on the mixed convection, which is presented by m, the power index of the power-law form heating condition. Numerical results for both heating conditions are carried out. Distributions of dimensionless temperature and velocity profiles for both Darcy and non-Darcy models are presented. Received on 26 May 1997     

Numerical simulation (DNS and LES) of manipulated turbulent boundary layer flow over a surface-mounted fence

Results from numerical simulations are presented for manipulated turbulent boundary layer flow over a surface-mounted fence, for a Reynolds number of Re_h=3000 (based on fence height, h , and maximum inflow velocity, U_∞ ). First, a reference data set was provided from a Direct Numerical Simulation (DNS) using 51.6 million grid points to resolve all the relevant spatial scales of the flow. A Large-Eddy Simulation (LES), using 1.67 million grid points, was validated with this reference solution and compared with experimental data for the same Reynolds number. Then, manipulated flow cases were investigated applying time-periodic forcing through a narrow slot upstream of the flow obstacle. High-frequency forcing, with Str₁=f₁h/U_∞=0.60 , leads to about 10% reduction of the mean re-attachment length. A much stronger reduction of about 36% could be achieved by low-frequency forcing with Str₂=f₂h/U_∞=0.08 . In the latter case, large-scale coherent structures are created between the location of the disturbance and the fence, they roll over the flow obstacle (nearly unaffected) and in rolling downstream they still grow in size until they fill out the entire height of the separation zone behind the fence. In agreement with corresponding experiments of Siller and Fernholz in 1997 for a higher Reynolds number ( Re_h=10500 ) the optimum forcing Strouhal number seems to be related to the low-frequency movement of the entire separation bubble and not to the instability mode of the separating shear layer.©2000 Éditions scientifiques et médicales Elsevier SAS     

Heat and mass transfer characteristics of the self-similar boundary-layer flows induced by continuous surfaces stretched with rapidly decreasing velocities

The mechanical and thermal characteristics of the self-similar boundary-layer flows induced by continuous surfaces stretched with rapidly decreasing power-law velocities U _w∝x ^m , m<?1 are considered. Comparing to the well studied cases of the increasing stretching velocities (m>0) several new features of basic significance have been found. Thus: (i) for m<?1 the boundary layer equations admit self-similar solutions only if a lateral suction is applied; (ii) the dimensionless suction velocity f _w<0 must be strong enough, i.e. f _w<f _w,max(m) where f _w,max(m) depends on m so that its absolute maximum max (f _w,max(m))=?2.279 is reached for m→?∞, while for m→?1, f _w,max(m)→?∞; (iii) the case {m→?∞, f _w,max(m)=?2.279} of the flow boundary value problem is isomorphic to the stretching problems with exponentially decreasing velocities U _w∝e ^ax with arbitrary a<0; (iv) for any fixed m<?1 and f _w<f _w,max(m) the flow problem admits a non-denumerable infinity of multiple solutions corresponding to the values of the dimensionless skin friction f ^″(0)≡s belonging to a finite interval s∈ [s _min(f _w,m), s _max(f _w,m)]; (v) the solution is only unique for f _w=f _w,max(m) where s=s _min(f _w,m)= s _max(f _w,m) holds; (vi) to every one of the multiple solutions of the flow problem there corresponds a unique solution of the heat transfer problem with a wall temperature distribution T _w?T _∞∝x ⁿ and a well defined and distinct value of the dimensionless wall temperature gradient ?^′(0), except for the cases n=(|m|?1)/2 where ?^′(0) has the same value ?^′(0)=Pr·f _w for the whole class of flow solutions with s∈[s _min(f _w,m), s _max(f _w,m)]; (vii) for f _w→?∞ one obtains the `asymptotic suction profiles' corresponding to s=s _min(f _w,m)?f _w and ?^′(0)?Pr·f _w in an explicit analytic form. The paper includes several examples which illustrate the dependence of the heat and fluid flows induced by surfaces stretching with rapidly decreasing velocities on the physical parameters f _w, m, n and Pr.     

The Effect of Reynolds Number on the Passive Scalar Field in the Turbulent Wake of a Circular Cylinder

In this paper we investigate by experiments the effect of Reynolds number on a passive scalar (temperature) field in the turbulent wake of a slightly heated circular cylinder. The Reynolds number defined by Re≡U _∞ d/ν (see Nomenclature) is varied from Re= 1200 to Re= 8600. Temperature differential above ambient is chosen to be the passive scalar quantity. Present measurements are conducted using a cold wire (0.63 μm) probe. Results obtained suggest that Reynolds number in general has significant influence on the scalar mixing characteristics in the entire wake flow. Specifically, as Re increases, the mean scalar spreads out more rapidly, the scalar fluctuation intensity increases; however, its variance decays at a lower rate with downstream distance. It is also found that an increase of Re accelerates the streamwise evolution of the scalar probability density function from highly non-Gaussianity to near Gaussianity along the wake centreline. This reflects the reduction in length of the Karman-vortex street caused by an increase of Re.     

W2,p Estimates for the Parabolic¶Monge-Ampère Equation

When u is a solution to the equation ?u _t det D _x ² u=f with f positive, continuous, and f _t satisfying certain growth conditions, we establish estimates in L ^∞ for u _t and show that D _x ² u satisfies uniform interior estimates in L ^p for 0相似文献   

PIV measurements of the wake behind a rotationally oscillating circular cylinder

The near-wake behind a circular cylinder undergoing rotational oscillatory motion with a relatively high forcing frequency has been investigated experimentally. Experiments were carried out varying the ratio of the forcing frequency f_f to the natural vortex shedding frequency f_n in the range of 0.0 (stationary) to 1.6 at an oscillation amplitude of θ_A=30° and Reynolds number of Re=4.14×10³. Depending on the frequency ratio (F_R=f_f /f_n), the near-wake flow could be divided into three regimes—non-lock-on (F_R=0.4), transition (F_R=0.8, 1.6) and lock-on (F_R=1.0) regimes—with markedly different flow structures. When the frequency ratio was less than 1.0 (F_R⩽1.0), the rotational oscillatory motion of the cylinder decreased the length of the vortex formation region and enhanced the mutual interaction between large-scale vortices across the wake centerline. The entrainment of ambient fluid seemed to play an important role in controlling the near-wake flow and shear-layer instability. In addition, strong vortex motion was observed throughout the near-wake region. The flow characteristics changed markedly beyond the lock-on flow regime (F_R=1.0) due to the high frequency forcing. At F_R=1.6, the high frequency forcing decreased the size of the large-scale vortices by suppressing the lateral extent of the wake. In addition, the interactions between the vortices shed from both sides of the cylinder were not so strong at this forcing frequency. As a consequence, the flow entrainment and momentum transfer into the wake center region were reduced. The turbulent kinetic energy was large in the region near the edge of the recirculation region, where the vortices shed from both sides of the cylinder cross the wake centerline for all frequency ratios except for the case of F_R=1.6. The temporally resolved quantitative flow information extracted in the present work is useful for understanding the effects of open-loop active flow control on the near-wake flow structure.     

The effect of suction or injection on the boundary layer flows induced by continuous surfaces stretched with prescribed skin friction

The boundary layer flow and heat transfer on a stretched surface moving with prescribed skin friction is studied for permeable surface. Three major cases are studied for isothermal surface (n=0) stretched corresponding to different dimensional skin friction boundary conditions namely; skin friction at the surface scales as (x ^?1/2) at m=0, constant skin friction at m=1/3 and skin friction scales as (x) at m=1. The constants m and n are the indices of the power law velocity and temperature exponent respectively. Similarity solutions are obtained for the boundary layer equations subject to power law temperature and velocity variation. The effect of various governing parameters, such as Prandtl number Pr, suction/injection parameter f _w , m and n are studied. The results show that for isothermal surface increasing m enhances the dimensionless heat transfer coefficient for fixed f _w at the suction case and the reverse is true at the injection case. Furthermore, for fixed m, as f _w increases the dimensionless heat transfer coefficient increases. Large enhancements are observed in the heat transfer coefficient as the temperature boundary condition along the surface changes from uniform to linear where the dimensional skin friction is of order (x) at m=1. This enhancement decreases as the suction increases.     

Effect of the shape of a body on the intermittency of turbulent flow in an axisymmetrical wake

Experimental data regarding the distribution of the intermittency coefficient of the turbulent flow of an incompressible liquid in the axisymtnetrical wakes behind a sphere or a solid of revolution with an elongation of 8∶1 are presented. The original measurements were carried out at a Reynolds number of Re=U_∞D/v=10⁴ (U_∞) is the velocity of the incident flow, D is the diameter of the middle cross section). The shape of the solid is shown to have a considerable effect on the form of intermittency in the far automodel wake.     

Flow-induced vibrations of two circular cylinders in tandem with shear flow at low Reynolds number

The flow-induced vibrations of two elastically mounted circular cylinders subjected to the planar shear flow in tandem arrangement are studied numerically at Re=160. A four-step semi-implicit Characteristic-based split (4-SICBS) finite element method is developed under the framework of the fractional step method to cope with the vortex-induced vibration (VIV) problem. For the computational code verification, two benchmark problems are examined in the laminar region: flow-induced vibration of an elastically mounted cylinder having two degrees of freedom and past two stationary ones in tandem arrangement. Regarding the two-cylinder VIVs in shear flow, the computation is conducted with the cylinder reduced mass M_r=2.5π and the structural damping ratio ξ=0.0. The effects of some key parameters, such as shear rate (k=0.0, 0.05, 0.1), reduced velocity (U_r=3.0–18.0) and spacing ratio (L_x/D=2.5, 3.5, 4.5, 8.0), are demonstrated. It is observed that the shear rate and reduced velocity play an important role in the VIVs of both cylinders at various center-to-center distances. Additionally, in comparison with the single cylinder case, a further study indicated that the gap flow has a significant impact on such a dynamic system, leading it to be more complex. The results show that, the performances of ‘dual-resonant’ are discovered in the shear flow. A valley is formed in transverse oscillation amplitude of DC for each spacing ratio when U_r is about 6.0. For the X–Y trajectories of the circular cylinders, figure-eight, figure-O and oval shape are obtained. Finally, the interactions between cylinders are revealed, together with the wake-induced vibration (WIV) mechanism underlying the oscillation characteristics of both cylinders exposed to shear flow. Besides, the “T+P” wake pattern is discovered herein.     

Influence of Variable Permeability on Combined Convection Along a Nonisothermal Wedge in a Saturated Porous Medium

Mixed convection along a vertical nonisothermal wedge embedded in a fluid-saturated porous media incorporating the variation of permeability and thermal conductivity is studied. The surface temperature is assumed to vary as a power of the axial coordinate measured from the leading edge of the plate. A nonsimilar mixed convection parameter and a pseudo-similarity variable are introduced to cast the governing boundary layer equations into a system of dimensionless equations which are solved numerically using finite difference method. The entire mixed convection regime is covered by the single nonsimilarity parameter =[1+(Ra _x /Pe _x )^1/2]^–1 from pure forced convection (=1) to pure free convection (=0). The problem is solved using nonsimilarity solution for the case of variable wall temperature. Velocity and temperature profiles as well as local Nusselt number are presented. The wedge angle geometry parameter is ranged from 0 to 1.     

Unsteady separated and reattaching turbulent flow over a two-dimensional square rib

The spatio-temporal characteristics of the separated and reattaching turbulent flow over a two-dimensional square rib were studied experimentally. Synchronized measurements of wall-pressure fluctuations and velocity fluctuations were made using a microphone array and a split-fiber film, respectively. Profiles of time-averaged streamwise velocity and wall-pressure fluctuations showed that the shear layer separated from the leading edge of the rib sweeps past the rib and directly reattaches on the bottom wall (x/H=9.75) downstream of the rib. A thin region of reverse flow was formed above the rib. The shedding large-scale vortical structures (fH/U₀=0.03) and the flapping separation bubble (fH/U₀=0.0075) could be discerned in the wall-pressure spectra. A multi-resolution analysis based on the maximum overlap discrete wavelet transform (MODWT) was performed to extract the intermittent events associated with the shedding large-scale vortical structures and the flapping separation bubble. The convective dynamics of the large-scale vortical structures were analyzed in terms of the autocorrelation of the continuous wavelet-transformed wall pressure, cross-correlation of the wall-pressure fluctuations, and the cross-correlation between the wall pressure at the time-averaged reattachment point and the streamwise velocity field. The convection speeds of the large-scale vortical structures before and after the reattachment point were U_c=0.35U₀ and 0.45U₀, respectively. The flapping motion of the separation bubble was analyzed in terms of the conditionally averaged reverse-flow intermittency near the wall region. The instantaneous reattachment point in response to the flapping motion was obtained; these findings established that the reattachment zone was a 1.2H-long region centered at x/H=9.75. The reverse-flow intermittency in one period of the flapping motion demonstrated that the thin reverse flow above the rib is influenced by the flapping motion of the separation bubble behind the rib.     

Prediction of stagnation point heat transfer for a slot jet impinging on a flat surface

The fluid flow and heat transfer for a slot jet impinging on a flat plate has been analysed for different nozzle-to-plate spacing. The available potential flow solution has been used to solve the boundary layer and energy equations by using the Blasius-Frossling series solution method. The friction factor and Nusselt number have been evaluated as a function of the dimensionless distance from the stagnation point. Correlation for the Stanton number at the Stagnation point, is obtained in terms of velocity gradient at the stagnation point and Reynolds number.

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Berechnung des Wärmeübergangs am Staupunkt für einen Strahl, der senkrecht auf eine ebene Fläche trifft

Zusammenfassung Für einen Fluidstrahl, der senkrecht auf eine ebene Platte trifft, wurden für verschiedene Anordnungen von Düse und Platte Strömung und Wärmeübertragung untersucht. Die beschreibende Potentialtheorie wurde verwendet, um die Grenzschicht und Energiegleichungen mit Hilfe der Blasius-Frossling-Reihenentwicklung zu lösen. Reibungsfaktor und Nusseltzahl sind als eine Funktion des dimensionslosen Abstandes vom Staupunkt dargestellt. Die Beziehung für die Stanton-Zahl am Staupunkt ist in den Ausdrücken des Geschwindigkeitsgradienten am Staupunkt und der Reynoldszahl enthalten.

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Nomenclature A ₁ dimensionless coefficient - a dimensionless parameter - b dimensionless parameter - C _f friction factor,C _f= ₀/(1/2w ² ) - C _p specific heat at constant pressure - F ₀ function ofPr and - G ₄ function ofPr and - f ₁ function of - h heat transfer coefficient - k thermal conductivity - l half-width of slot nozzle - Nu Nusselt number,Nu=hl/k - Pr Prandtl number,Pr=v/ - Re Reynolds number,Re=w l/v - St Stanton number,St=Nu/(Re · Pr) - t temperature - t _w wall temperature - t ambient temperature - U dimensionless velocity,U=u/w - U _f dimensionless free-stream velocity,U _f =u _f /w - U _s dimensionless mainstream velocity along the plate,U _s =u _s /w - u velocity component inx-direction - u _f free stream velocity - u _s mainstream velocity along the plate - w velocity component inz-direction - w velocity at the nozzle exit - x coordination along the plate - X dimensionless distance from the stagnation point along the plate,X=x/l - Y ratio ofU _s andU _f ,Y=U _s /U _f - z coordinate perpendicular to the plate - z _n height of the nozzle above the plate - Z dimensionless height of the nozzle above the plate,Z=z _n /l - thermal diffusivity,=k/( C _p) - dimensionless parameter - dimensionless coordinate perpendicular to the plate - viscosity - kinematic viscosity - ₀ shear stress at the wall - stream function     

Wavefronts for a cooperative tridiagonal system of differential equations

Consider the infinite system of nonlinear differential equations \(\dot u\) _n =f(_n?1, u_n, u_n+1),nε?, wherefεC ¹,D ₁ f > 0,D ₃ f>0, andf(0, 0, 0) = 0 =f(1, 1, 1). Existence of wavefronts—i.e., solutions of the formu _n (t) = U(n + ct), wherecε?,U(? ∞) = 0,U(+∞) = 1, andU is strictly increasing—is shown for functionsf which satisfy the condition: there existsa, 0<a<1, such thatf(x, x,x)<0 for 0<x<a andf(x, x, x) > 0 fora < x < 1.     

Mixed convection boundary layer flow over a horizontal circular cylinder with Newtonian heating

The steady mixed convection boundary layer flow over a horizontal circular cylinder, generated by Newtonian heating in which the heat transfer from the surface is proportional to the local surface temperature, is considered in this study. The governing boundary layer equations are first transformed into a system of non-dimensional equations via the non-dimensional variables, and then into non-similar equations before they are solved numerically using a numerical scheme known as the Keller-box method. Numerical solutions are obtained for the skin friction coefficient Re ^1/2 C _f and the local wall temperature θ _w (x) as well as the velocity and temperature profiles with two parameters, namely the mixed convection parameter λ and the Prandtl number Pr.     

Reynolds' analogy for the thermal convection driven by nonisothermal stretching surfaces

In the present paper the steady boundary-layer flows induced by permeable stretching surfaces with variable temperature distribution are investigated under the aspect of Reynolds' analogy r = St _x /C _f(x). It is shown that for certain stretching velocities and wall temperature distributions, “Reynolds' function”r, i.e. the ratio of the local Stanton number St _x and the skin friction coefficient C _f(x) equals −1/2 for any value of the Prandtl number Pr and of the dimensionless suction/injection velocity f _w. In all of these cases, the dimensionless temperature field ϑ is connected to the dimensionless downstream velocity f ^′ by the simple relationship ϑ=(f ^′)^Pr. It is also shown that in the general case, Reynolds' function r may possess several singularities in f _w. The largest of them represents a critical value, so that for f _w<f _w,crit the solutions of the energy equation (although they still satisfy all the boundary conditions) become nonphysical.     

The buoyancy effects on the boundary layers induced by continuous surfaces stretched with rapidly decreasing velocities

Heat and mass transfer characteristics of the self-similar boundary layer flows induced by continuous surfaces stretched with rapidly decreasing power law velocities U_w x^m, m < –1 are considered for mixed convection flow. The effect of various governing parameters, such as Prandtl number Pr, temperature exponent n, dimensionless injection/suction velocity f_w, and the mixed convection parameter = s Gr/Re² are studied. These parameters have great effects on velocity and temperature profiles, heat transfer coefficient, and skin friction coefficient at the moving surface. Results show that similarity solutions exist only when the condition n = 2m – 1 is satisfied. Critical values of , Nu/Re^0.5 and C_f Re^0.5 are obtained for predominate natural convection for different Prandtl numbers at m = –2, –6 and n = –5, and –13 respectively. Results also show that the effect of buoyancy is more significant for weak than for strong suction. Furthermore, critical Prandtl numbers where f_w profiles have minimums are obtained for m = –2 and –6. Finally, critical values of , C_f Re^0.5 are also obtained for predominate natural convection for both m = –2 and –6.     

Finite difference analysis of unsteady natural convection flow along an inclined plate with variable surface temperature and mass diffusion

An analysis is performed to study the transient laminar natural convection flows along an inclined semi-infinite flat plate in which the wall temperatureT _w ^′ and species concentration on the wallC _w ^′ vary as the power of the axial co-ordinate in the formT _w ^′ (x)=T _∞ ^′ +ax ⁿ andC _w ^′ =C _∞ ^′ +bx ^m respectively. The dimensionless governing equations considered here are unsteady, two-dimensional, coupled and non-linear integro-differential equations. A finite difference scheme of Crank-Nicolson type is employed to solve the problem. The velocity, temperature, concentration, skin friction, Nusselt number and Sherwood number are studied in detail for various sets of values of parameters. Correlation equations are also established for Nusselt number and Sherwood number in terms of parameters.     

Passive Scalar Transport in a Turbulent Cylinder Wake in the Presence of a Downstream Cylinder

This work aims to investigate how the presence of a downstream cylinder affects the passive scalar transport in a cylinder wake. The wake was generated by two tandem brass circular cylinders of the same diameter (d). The cylinder centre-to-centre spacing L/d was 1.3, 2.5 and 4.0, respectively, covering the three typical flow regimes of this flow. The upstream cylinder was slightly heated. Measurements were conducted at x/d= 10 and Re (≡ dU _∞/ν, where U _∞ is the free-stream velocity and ν is the kinematic viscosity of fluid) = 7000. A three-wire probe consisting of an X-wire and a cold wire was used to measure the velocity and temperature fluctuations, while an X-wire provided a phase reference. The phase-averaged velocity vectors and vorticity display single vortex street behind the downstream cylinder, irrespective of the flow regimes. However, the detailed flow structure exhibits strong dependence on L/d in terms of the Strouhal number, the vortex strength and its downstream evolution. This naturally affects passive scalar transport. The coherent and incoherent heat flux vectors show significant variation for different L/d.     

Forced Convection Heat Transfer from a Heated Cylinder in an Axial Background Flow in a Porous Medium: Three Exactly Solvable Cases

Assuming a background flow of velocity U = U(x) in the axial direction x of a circular cylinder with surface temperature distribution T _w = T _w (x) in a saturated porous medium, for the temperature boundary layer occurring on the cylinder three exactly solvable cases are identified. The functions {U(x), T _w (x)} associated with these cases are given explicitly, and the corresponding exact solutions are expressed in terms of the modified Bessel function K ₀ (z), the incomplete Gamma function Γ (a, z) and the confluent hypergeometric function U(a, b, z), respectively. The correlation between the Nusselt number and the Péclet number as well as the curvature effects on the heat transfer are discussed in all these cases in detail. Some “universal” features of the exponential surface temperature distribution are also pointed out.