The virtual origin of riblets in an open channel flow

In this paper, a method using the mean velocity profiles for the buffer layer was developed for the estimation of the virtual origin over a riblets surface in an open channel flow. First, the standardized profiles of the mixing length were estimated from the velocity measurement in the inner layer, and the location of the edge of the viscous layer was obtained. Then, the virtual origins were estimated by the best match between the measured velocity profile and the equations of the velocity profile derived from the mixing length profiles. It was made clear that the virtual origin and the thickness of the viscous layer are the function of the roughness Reynolds number. The drag variation coincided well with other results.Nomenclature f _r skin friction coefficient - f _ro skin friction coefficient in smooth channel at the same flow quantity and the same energy slope - g gravity acceleration - H water depth from virtual origin to water surface - H ⁺ u*H/ - H false water depth from top of riblets to water surface - H ⁺ u*H/ - I _e streamwise energy slope - I _b bed slope - k riblet height - k ⁺ u*k/ - l mixing length - l _s standardized mixing length - Q flow quantity - Re Reynolds number volume flow/unit width/v - s riblet spacing - u mean velocity - u* friction velocity = - u* false friction velocity = - y distance from virtual origin - y distance from top of riblet - y ₀ distance from top of riblet to virtual origin - y _v distance from top of riblet to edge of viscous layer - y ⁺ u*y/ - y ⁺ u*y/ - y ₀ ⁺ u*y ₀/ - u ⁺ u*y/ - shifting coefficient for standardization - thickness of viscous layer=y ₀+y - ⁺ u*/ - ⁺ u*/ - eddy viscosity - ridge angle - v kinematic viscosity - density - shear stress     

Mean and turbulent velocity measurements of supersonic mixing layers

The behavior of supersonic mixing layers under three conditions has been examined by schlieren photography and laser Doppler velocimetry. In the schlieren photographs, some large-scale, repetitive patterns were observed within the mixing layer; however, these structures do not appear to dominate the mixing layer character under the present flow conditions. It was found that higher levels of secondary freestream turbulence did not increase the peak turbulence intensity observed within the mixing layer, but slightly increased the growth rate. Higher levels of freestream turbulence also reduced the axial distance required for development of the mean velocity. At higher convective Mach numbers, the mixing layer growth rate was found to be smaller than that of an incompressible mixing layer at the same velocity and freestream density ratio. The increase in convective Mach number also caused a decrease in the turbulence intensity ( _u/U).List of symbols a speed of sound - b total mixing layer thickness between U ₁ – 0.1 U and U ₂ + 0.1 U - f normalized third moment of u-velocity, f u³/(U)³ - g normalized triple product of u² , g u²/(U)³ - h normalized triple product of u ², h u ²/(U)³ - l _u axial distance for similarity in the mean velocity - l _u axial distance for similarity in the turbulence intensity - M Mach number - M _c convective Mach number (for ₁ = ₂), M _c (U ₁ – U ₂)/(a ₁ + a ₂) - P static pressure - r freestream velocity ratio, r U ₂/U ₁ - Re unit Reynolds number, Re U/ - s freestream density ratio, s ₂/₁ - T _t total temperature - u instantaneous streamwise velocity - u deviation of u-velocity, uu – U - U local mean streamwise velocity - U ₁ primary freestream velocity - U ₂ secondary freestream velocity - average of freestream velocities, (U ₁ + U ₂)/2 - U freestream velocity difference, U U ₁ – U ₂ - instantaneous transverse velocity - v deviation of -velocity, – V - V local mean transverse velocity - x streamwise coordinate - y transverse coordinate - y ₀ transverse location of the mixing layer centerline - ensemble average - ratio of specific heats - boundary layer thickness (y-location at 99.5% of free-stream velocity) - similarity coordinate, (y – y ₀)/b - compressible boundary layer momentum thickness - viscosity - density - standard deviation - dimensionless velocity, (U – U ₂)/U - 1 primary stream - 2 secondary stream A version of this paper was presented at the 11th Symposium on Turbulence, October 17–19, 1988, University of Missouri-Rolla     

Prediction of stagnation point heat transfer for a single round jet impinging on a concave hemispherical surface

This paper deals with a systematic procedure for assessment of fluid flow and heat transfer parameters for a single round jet impinging on a concave hemispherical surface. Based on Scholkemeier's modifications of the Karman-Pohlhausen integral method, expressions are derived for evaluation of the momentum thickness, boundary layer thickness and the displacement thickness at the stagnation point. This is followed by the estimation of thermal boundary layer thickness and local heat transfer coefficients. A correlation is presented for the Nusselt number at the stagnation point as a function of the Reynolds number for different non-dimensional distances from the exit plane of the jet to the impingement surface.

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Bestimmung des Staupunktes bei der Wärmeübertragung für einen einzelnen Strahl, der auf eine konkave halbkugelige Oberfläche trifft

Zusammenfassung Diese Arbeit beschäftigt sich mit dem systematischen Verfahren der Bewertung von Fluidströmungen und Wärmeübertragungsparametern für einen einzelnen runden Strahl, der auf eine konkave halbkugelförmige Oberfläche trifft. Das Verfahren beruht auf Scholkemeiers Modifikation des Karman-Pohlhausen Integrationsverfahrens. Ausdrücke sind für die Berechnung der Impuls-Dicke, der Grenzschichtdicke und der Verschiebungsdicke am Staupunkt hergeleitet worden. Dies ist aus der Berechnung der thermischen Grenzschichtdicke und des lokalen Wärmeübertragungskoeffizienten abgeleitet worden. Es wird eine Gleichung für die Nusselt-Zahl am Staupunkt als Funktion der Reynolds-Zahl für verschiedene dimensionslose Abstände vom Strahlaustrittspunkt bis zum Auftreffpunkt auf die Oberfläche vorgestellt.

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Nomenclature c _p specific heat at constant pressure - d diameter of single round nozzle - h ₀ heat transfer coefficient at the stagnation point - H distance from the exit plane of the jet to the impingement surface - k thermal conductivity - Nu _0.5 Nusselt number based on impinging jet quantities=h _0.50/k - Nu _{0.5, 0} stagnation point Nusselt number=h _{0 0,50}/k - p pressure - p _a ambient pressure - p ₀ maximum pressure or stagnation pressure - p(x) static pressure at a distancex from the stagnation point - R radius of curvature of the hemisphere - Re _J jet Reynolds number=U _Jd/ - Re _0.5 Reynolds number based on impinging jet quantities=u _{m0 0.50}/ - T temperature - T _a room temperature - T _J jet temperature - T _W wall temperature - u velocity component inx andx directions (Fig. 1) - u _m jet centerline (or maximum) free jet velocity: external (or maximum) boundary layer velocity aty= _m - u _m0 arrival velocity defined as the maximum velocity the free jet would have at the plane of impingement if the plane were not there - U _J jet exit velocity - x* non-dimensional coordinate starting at the stagnation point=x/2 _0.50 - x, y rectangular Cartesian coordinates - y coordinate normal to the wall starting at the wall - ratio of thermal to velocity boundary layer thickness= _T/_m - ₀ ratio of thermal to velocity boundary layer thickness at the stagnation point - * inner layer displacement thickness - _0.50 jet half width at the plane of impingement if the plate were not there - _m inner boundary layer thickness atu=u _m - Pohlhausen's form parameter - dynamic viscosity - kinematic viscosity=/ - fluid density - momentum thickness - ₀ momentum thickness at the stagnation point     

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

A systematic procedure has been laid out for assessment of fluid flow and heat transfer parameters for a slot jet impinging on a concave semicylindrical surface. Based on Walz's modifications of the Karman-Pohlhausen integral method, expressions have been derived for evaluation of the momentum thickness, boundary layer thickness and the displacement thickness at the stagnation point. The work then has been extended for the estimation of thermal boundary layer thickness and local heat transfer coefficients. A correlation has been presented for the Nusselt number at the stagnation point as a function of the Reynolds number for different non-dimensional distances from the exit plane of the jet to the impingement surface.

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Berechnung des Wärmeübergangs im Staupunkt eines Strahles, der aus einer rechteckigen öffnung auf eine konkave halbzylindrische Fläche auftrifft

Zusammenfassung Es wurde eine systematische Prozedur für die Abschätzung von Strömungs- und Wärmeübergangsparametern für einen Strahl, der auf eine konkave halbzylindrische Fläche auftrifft, aufgestellt. Basierend auf Walz's Modifikationen der Karman-Pohlhausen Integral-Methode, wurden Ausdrücke für die Berechnung der Impulsdicke, der Grenzschichtdicke und die Versetzungsdicke am Staupunkt abgeleitet. Die Arbeit wurde dann auf die Abschätzung der thermischen Grenzschichtdicke und der lokalen Wärmeübertragungskoeffizienten ausgedehnt. Es wird eine Beziehung für die Nusselt-Zahl am Staupunkt als eine Funktion der Reynolds-Zahl für verschiedene dimensionslose Abstände von der Austrittsfläche des Schlitzes bis zur Aufprallfläche aufgestellt.

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Nomenclature c _p specific heat at constant pressure - h ₀ heat transfer coefficient at the stagnation point - H distance from the exit plane of the jet to the impingement surface - k thermal conductivity - Nu _.5 Nusselt number based on impinging jet quantities =h _0.50/k - Nu _.5,0 stagnation point Nusselt number =h _{0 0.50}/k - p pressure - p _a ambient pressure - p ₀ maximum pressure or stagnation pressure - p(x) static pressure at a distancex from the stagnation point - p(x*) static pressure at nondimensional distancex* from the stagnation point - Re _J jet Reynolds number =U _J W/ - Re _0.5 Reynolds number based on impinging jet quantities =u _{m0 0.50}/ - T temperature - T* nondimensional temperature =(T–T _W)/(T _J–T _W) - T _a room temperature - T _J jet temperature - T _W wall temperature - u velocity component inx andx directions - u _m jet centerline (or maximum) free jet velocity: external (or maximum) boundary layer velocity aty = _m - u _m0 arrival velocity defined as the maximum velocity the free jet would have at the plane of impingement if the plane were not there - U _J jet exit velocity - W jet nozzle width - x* nondimensional coordinate starting at the stagnation point =x/2 _0.50 - x, y rectangular cartesian coordinates - y coordinate normal to the wall and starting at the wall - ratio of thermal to velocity boundary layer thickness = _T/ _m - ₀ ratio of thermal to velocity boundary layer thickness at the stagnation point - * inner layer displacement thickness - _.50 jet half width at the plane of impingement if the plate were not there - d_.5 free jet (half width) thickness whereu=u _m/2 - _m inner boundary layer thickness atu =u _m - _T thermal boundary layer thickness - nondimensional coordinate normal to wall =y/ _m - _T nondimensional coordinate normal to wall =y/ _T - Pohlhausen's form parameter - dynamic viscosity - kinematic viscosity = / - fluid density - momentum thickness - ₀ momentum thickness at the stagnation point     

An experimental study on the two-dimensional opposed wall jet in a turbulent boundary layer: Change in the flow pattern with velocity ratio

Results of the measurement of flow properties in a two-dimensional turbulent wall jet which is injected into the turbulent boundary layer in the direction opposite to that of the boundary layer flow are presented by varying the ratio of the jet issuing velocity to the mainstream velocity . This flow includes the flow separation and the recirculating flow, and so it requires the magnitude and direction of instantaneous velocity be measured. In the present work, a tandem hot-wire probe is manufactured and its characteristics are examined experimentally. With the use of this probe the change in the penetration length of the jet with the velocity ratio is investigated. It is clarified that two regimes of flow patterns exist: in the low velocity ratio the penetration length of the jet changes little with , and in the high velocity ratio it goes far from the nozzle with increasing . Streamlines, turbulence intensity contours and static pressure contours are presented in the two typical velocity ratios corresponding to each of two flow patterns, and they are compared.List of symbols b ₀ nozzle width (Fig. 1) - , e mean and fluctuating output voltages of hot-wire anemometer - p, p mean static pressure, p = p – p _o - p ₀ static pressure in undisturbed mainstream - p _w , p _w mean wall pressure, p _w = p _w – p _o - U ₀ mainstream velocity - U _j jet velocity at the nozzle exit - , u mean and fluctuating velocity components in x-direction - u _e effective cooling velocity - x distance along the wall from nozzle exit - x _pmax, x _pmin positions where the wall pressure has maximum and minimum values respectively - x _s penetration length of jet - y distance from the wall - forward flow fraction - ₁, ₂ yaw and pitch angles of flow direction (Fig. 4) - velocity ratio, = U _j /U _o - density of the fluid - nondimensional stream function The authors wish to express their appreciation to Prof. Toshio Tanaka of Gifu University for his advice in the course of the experiment. They also would like to thank the Research Foundation for the Electrotechnology of Chubu which partly supported this work.     

Bursting and structure of the turbulence in an internal flow manipulated by riblets

The buffer layer of an internal flow manipulated by riblets is investigated. The distributions of the ejection and bursting frequency from the beginning to the middle part of the buffer layer, together with high moments of the fluctuating streamwise velocity,u, and its time derivative are reported. The profiles of the ejection and bursting frequency are determined and compared using three single point detection schemes. The effect of the riblets on the bursting mechanism is found confined in a localized region in the buffer layer. The multiple ejection bursts are more affected than the single ejection bursts. The skewness and flatness factors of theu signal are larger in the manipulated layer than in the standard boundary layer. That, also holds true for the flatness factor of the time derivative, but the Taylor and Liepman scales are not affected. The spectrum of theu signal is altered at the beginning part of the viscous sublayer.Nomenclature u Friction velocity - Viscosity - l _v ;f _v wall scalesv/u ;u ² /v - y Vertical distance to the wall - z Spanwise extent - (⁺) Variable normalized with wall scales - u Velocity;u=Turbulence intensity - h, s Height and width of the riblets - f _e Ejection frequency - f _b Bursting frequency - f _BME Frequency of the Bursts with Multiple Ejection - f _BSE Frequency of Single Ejection Bursts - S andS _du/dt Skewness factor ofu and its time derivative - F _u andF _du/dt Flatness factor ofu and its time derivative Abbreviations SBL Standard (non-manipulated) Boundary Layer - MBL Manipulated Boundary Layer - BME Bursts with Multiple Ejections - BSE Bursts with Single Ejections - VITA Variable Interval Time Averaging technique - u–l u-level technique - mu Modifiedu-level technique     

Experimental investigation of mean flow characteristics of slot jet impingement on a cylinder

An experimental investigation is made to study the flow characteristics of slot jet impingement on a cylinder. The velocity profiles and pressure distribution around the cylinder are reported for various parameters namely, the flow rate, width of the nozzle, distance of the cylinder from the jet exit and eccentricity of the cylinder to the jet axis.

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Experimentelle Untersuchung über die Strömungseigenschaften eines Düsenstrahls, der auf einen Zylinder aufprallt

Zusammenfassung Es wurde eine experimentelle Untersuchung gemacht, um die Strömungseigenschaften eines Düsenstrahls zu unterschen, der auf einen Zylinder prallt. Die Geschwindigkeitsprofile und die Druckverteilungen an dem Zylinder wurden für unterschiedliche Parameter dokumentiert. Die Parameter sind die Strömungsgeschwindigkeit, Düsengröße, Abstand zwischen Zylinder und Strahlaustritt und die Exzentrizität von Zylinder und Strahlachse.

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Nomenclature B breadth of the nozzle at the exit - D diameter of the cylinder - C _p pressure coefficient - g acceleration due to gravity - L distance of the cylinder from jet exit - P _a atmospheric pressure - P _c static pressure along the jet center-line - P ₀ stagnation pressure - P _W wall static pressure - Re _D Reynolds numberu _j D/ _a - Re _W Reynolds numberu _j W/ _a - r distance measured from cylinder surface in radial direction - r _m position of maximum velocity from cylinder surface - r _0.5 half width of the jet - u mean velocity - u _j mean velocity at the jet exit - u _m maximum velocity - W width of the nozzle - _a density of air - _m density of mercury - _w density of water - absolute viscosity - kinematic viscosity     

Similar solutions and the asymptotics of filtration equations

In this paper we consider the asymptotic behavior of solutions of the quasilinear equation of filtration as t. We prove that similar solutions of the equation u _t = (u )_xx asymptotically represent solutions of the Cauchy problem for the full equation u _t = [(u)]_xx if (u) is close to u for small u.     

Unsteady convective diffusion in a rotating parallel plate channel

The paper presents an exact analysis of the dispersion of a passive contaminant in a viscous fluid flowing in a parallel plate channel driven by a uniform pressure gradient. The channel rotates about an axis perpendicular to its walls with a uniform angular velocity resulting in a secondary flow. Using a generalized dispersion model which is valid for all time, we evaluate the longitudinal dispersion coefficientsK _i (i=1, 2, ...) as functions of time. It is shown thatK ₁=0 andK ₃,K ₄, ... decay rapidly in comparison withK ₂. ButK ₂ decreases with increasing (the dimensionless rotation parameter) for values of upto approximately =2.2. ThereafterK ₂ increases with further increase in and its value gets saturated for large values of (say, 500) and does not change any further with increase in . A physical explanation of this anomalous behaviour ofK ₂ is given.

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Instationäre konvektive Diffusion in einem rotierenden Parallelplattenkanal

Zusammenfassung In dieser Untersuchung wird eine exakte Analyse der Ausbreitung eines passiven Kontaminierungsstoffes in einer zähen Flüssigkeit gegeben, die, befördert durch einen gleichförmigen Druckgradienten, in einem Parallelplattenkanal strömt. Der Kanal rotiert mit gleichförmiger Winkelgeschwindigkeit um eine zu seinen Wänden senkrechte Achse, wodurch sich eine Sekundärströmung ausbildet. Unter Verwendung eines generalisierten, für alle Zeiten gültigen Dispersionsmodells werden die longitudinalen DispersionskoeffizientenK _i (i=1, 2, ...) als Funktionen der Zeit ermittelt. Es wird gezeigt, daßK ₁=0 gilt und dieK ₃,K ₄, ... gegenüberK ₂ schnell abnehmen.K ₂ nimmt ab, wenn , der dimensionslose Rotationsparameter, bis etwa zum Wert 2,2 ansteigt. Danach wächstK ₂ mit bis auf einem Endwert an, der etwa ab =500 erreicht wird. Dieses anomale Verhalten vonK ₂ findet eine physikalische Erklärung.

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List of symbols C solute concentration - D molecular diffusivity - K _i longitudinal dispersion coefficients - 2L depth of the channel - P ₀ dimensionless pressure gradient along main flow - Pe Péclet number - q velocity vector - Q _x,Q _y mass flux along the main flow and the secondary flow directions - dimensionless average velocity along the main flow direction - (x, y, z) Cartesian co-ordinates Greek symbols dimensionless rotation parameter - the inclination of side walls withx-axis - kinematic viscosity - fluid density - dimensionless time - angular velocity of the channel - dimensionless distance along the main flow direction - dimensionless distance along the vertical direction - dimensionless solute concentration - integral of the dispersion coefficientK ₂() over a time interval     

The effect of non-steady wall shear on pressure surges in conduits carrying viscous liquids

Summary A study is made of the attenuation of pressure surges in a two-dimension a channel carrying a viscous liquid when a valve at the downstream end is suddenly closed. The analysis differs from previous work in this area by taking into account the transient nature of the wall shear, which in the past has been assumed as equivalent to that existing in steady flow. For large values of the frictional resistance parameter the transient wall shear analysis results in less attenuation than given by the steady wall shear assumption.Nomenclature c /, ft/sec - e base of natural logarithms - F(x, y) integration function, equation (38) - (x) mean value of F(x, y) - g local acceleration of gravity, ft/sec² - h width of conduit, ft - k (2m–1)^{2 2} L/h ² c, equation (35) - k* 12L/h ² c, frictional resistance parameter, equation (46) - L length of conduit, ft - m positive integer - n positive integer - p pressure, lb/ft² - p ₀ constant pressure at inlet of conduit, lb/ft² - P pressure plus elevation head, p+gz, equation (4) - mean value of P over the conduit width h - P ₀ p ₀+gz ₀, lbs/ft² - R frictional resistance coefficient for steady state wall shear, lb sec/ft⁴ - s positive integer; also, condensation, equation (6) - t time, sec - t ct/L, dimensionless time - u x component of fluid velocity, ft/sec - u _m mean velocity in conduit, equation (12), ft/sec - u ₀(y) velocity profile in Poiseuille flow, equation (19), ft/sec - transformed velocity - U initial mean velocity in conduit - x distance along conduit, measured from valve (fig. 1), ft - x x/L, dimensionless distance - y distance normal to conduit wall (fig. 1), ft - y y/h, equation (25) - z elevation, measured from arbitrary datum, ft - z ₀ elevation of constant pressure source, ft - isothermal bulk compression modulus, lbs/ft² - _n , equation (37) - _n (2n–1)/2, equation (36) - viscosity, slugs/ft sec - / = kinematic viscosity, ft²/sec - density of fluid, slugs/ft³ - ₀ density of undisturbed fluid, slugs/ft³ - ø angle between conduit and vertical (fig. 1) The research upon which this paper is based was supported by a grant from the National Science Foundation.     

Deformation to breaking of deep water gravity waves

The results of laboratory observations of the deformation of deep water gravity waves leading to wave breaking are reported. The specially developed visualization technique which was used is described. A preliminary analysis of the results has led to similar conclusions than recently developed theories. As a main fact, the observed wave breaking appears as the result of, first, a modulational instability which causes the local wave steepness to approach a maximum and, second, a rapidly growing instability leading directly to the breaking.List of symbols L total wave length - H total wave height - crest elevation above still water level - trough depression below still water level - wave steepness =H/L - crest steepness =/L - trough steepness =/L - F ₁ forward horizontal length from zero-upcross point (A) to wave crest - F ₂ backward horizontal length from wave crest to zero-downcross point (B) - crest front steepness =/F ₁ - crest rear steepness =/F ₂ - vertical asymmetry factor=F ₂/F ₁ (describing the wave asymmetry with respect to a vertical axis through the wave crest) - µ horizontal asymmetry factor=/H (describing the wave asymmetry with respect to a horizontal axis: SWL) - T ₀ wavemaker period - L ₀ theoretical wave length of a small amplitude sinusoïdal wave generated at T _inf0 ^sup–1 frequency - ₀ average wave height     

Effect of the flow regime in an underexpanded jet issuing into a subsonic counterstream on the pressure distribution on a body

The pressure distribution on a cone with a half-angle =75°, from which a single central underexpanded jet issues into a subsonic counterstream, has been experimentally investigated. The effect of the flow regime in the jet on the pressure distribution is demonstrated. Generalized relations for the pressure on the body are obtained for various jet-flow momentum ratios J and flow Mach numbers M = 0.35–0.9; the Mach number M_a at the exit of the conical nozzle with half-angle _a=10° was equal to 2.9. The working medium of the jet and the flow was air with stagnation temperatures T_0a = T₀ 260–265°K. The ratio of the nozzle outlet radius to the radius of the maximum cross section of the cone _a/R_M=0.1.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 182–185, January-February, 1988.     

Schlieren interferometry in the mass diffusion of a two-dimensional jet

Objective of this work was to show that schlieren interferometry is a useful quantitative technique to study mass diffusion processes of binary gas mixtures and in particular to apply it to investigate the initial region of a two-dimensional helium jet mixing with air.For low Reynolds number (Re) helium mole fraction at the nozzle exit was lower than unity, and the jet initial region was absent. The latter was recovered at higher Re and the potential core length increased for increasing Re. Strong mole fraction gradients beside the nozzle lips were found for a distance of about one nozzle width.List of symbols d width of nozzle exit section - D mass diffusivity coefficient - f focal length of second lens - K Gladstone-Dale constant - i fringe order - n refractive index of gas - n _a refractive index of air - n =n – n _a - Re =Vd/v; Reynolds number - s distance between undisturbed fringes - s fringe displacement - Sc Schmidt number - u distance between test section and second lens - w distance between prism and focus of second lens - x distance from the plane of nozzle exit section - y distance from jet center plane - Z length of nozzle exit section - prism divergence angle - direction normal to undisturbed fringes - light wavelength - kinematic viscosity coefficient of helium - =n/(n _a – 1) - mass density - X mole fraction - X _cL mole fraction at nozzle centerplane     

A Strongly Degenerate Quasilinear Equation: the Parabolic Case

We prove the existence and uniqueness of entropy solutions of the Neumann problem for the quasilinear parabolic equation u_t=÷ a(u, Du), where a(z,)=f(z,), and f is a convex function of with linear growth as ||||, satisfying other additional assumptions. In particular, this class includes the case where f(z,)=(z)(), >0, and is a convex function with linear growth as ||||.     

Control of low-speed turbulent separated flow using jet vortex generators

A parametric study has been performed with jet vortex generators to determine their effectiveness in controlling flow separation associated with low-speed turbulent flow over a two-dimensional rearward-facing ramp. Results indicate that flow-separation control can be accomplished, with the level of control achieved being a function of jet speed, jet orientation (with respect to the free-stream direction), and jet location (distance from the separation region in the free-stream direction). Compared to slot blowing, jet vortex generators can provide an equivalent level of flow control over a larger spanwise region (for constant jet flow area and speed).Nomenclature C _p pressure coefficient, 2(P-P_)/V ² - C _Q total flow coefficient, Q/ v - D ₀ jet orifice diameter - Q total volumetric flow rate - R Reynolds number based on momentum thickness - u fluctuating velocity component in the free-stream (x) direction - V free-stream flow speed - VR ratio of jet speed to free-stream flow speed - x coordinate along the wall in the free-stream direction - jet inclination angle (angle between the jet axis and the wall) - jet azimuthal angle (angle between the jet axis and the free-stream direction in a horizontal plane) - boundary-layer thickness - momentum thickness - lateral distance between jet orifices A version of this paper was presented at the 12th Symposium on Turbulence, University of Missouri-Rolla, 24–26 Sept. 1990     

On nonlinear problems of mixed type: A qualitative theory using infinite-dimensional center manifolds

We consider the equation a(y)u_xx+div_y(b(y)_yu)+c(y)u=g(y, u) in the cylinder (–l,l)×, being elliptic where b(y)>0 and hyperbolic where b(y)<0. We construct self-adjoint realizations in L₂() of the operatorAu= (1/a) div_y(b_yu)+(c/a) in the case ofb changing sign. This leads to the abstract problem u_xx+Au=g(u), whereA has a spectrum extending to + as well as to –. For l= it is shown that all sufficiently small solutions lie on an infinite-dimensional center manifold and behave like those of a hyperbolic problem. Anx-independent cross-sectional integral E=E(u, u_x) is derived showing that all solutions on the center manifold remain bounded forx ±. For finitel, all small solutionsu are close to a solution on the center manifold such that u(x)-(x) Ce ^-(1-|x|) for allx, whereC and are independent ofu. Hence, the solutions are dominated by hyperbolic properties, except close to the terminal ends {±1}×, where boundary layers of elliptic type appear.     

Analysis of suddenly started laminar flow in the entrance region of a circular tube

Suddenly started laminar flow in the entrance region of a circular tube, with constant inlet velocity, is investigated analytically by using integral momentum approach. A closed form solution to the integral momentum equation is obtained by the method of characteristics to determine boundary layer thickness, entrance length, velocity profile, and pressure gradient.Nomenclature M(, , ) a function - N(, , ) a function - p pressure - p* p/1/2U ², dimensionless pressure - Q(, , ) a function - R radius of the tube - r radial distance - Re 2RU/, Reynolds number - t time - U inlet velocity, constant for all time, uniform over the cross section - u velocity in the boundary layer - u* u/U, dimensionless velocity - u ₁ velocity in the inviscid core - x axial distance - y distance perpendicular to the axis of the tube - y* y/R, dimensionless distance perpendicular to the axis - boundary layer thickness - * displacement thickness - /R, dimensionless boundary layer thickness - momentum thickness - absolute viscosity of the fluid - /, kinematic viscosity of the fluid - x/(R Re), dimensionless axial distance - density of the fluid - tU/(R Re), dimensionless time - _w wall shear stress     

Qualitative behavior of dissipative wave equations on bounded domains

The qualitative behavior of solutions of the mixed problem u_tt = u-a(x)u_t in IR x , u=0 on IR x , is studied in the case when a>0 and IRⁿ is bounded. Roughly speaking, if aa_min>0, then solutions decay at least as fast as exp t( –1/2a_min), with the possible exception of a finite dimensional set of smooth solutions whose existence is associated with a phenomenon of overdamping. If a_max is sufficiently small, depending on , then no overdamping occurs.Partially supported by NSF grant NSF GP 34260.This work was partially supported by the National Science Foundation under Grant No. GP 34260     

Mixed convection on vertical plate for fluids of any prandtl number

A mixed convection parameter=(Ra) ^1/4/(Re)^1/2, with=Pr/(1+Pr) and=Pr/(1 +Pr)^1/2, is proposed to replace the conventional Richardson number, Gr/Re², for combined forced and free convection flow on an isothermal vertical plate. This parameter can readily be reduced to the controlling parameters for the relative importance of the forced and the free convection,Ra ^1/4/(Re ^1/2 Pr ^1/3) forPr 1, and (RaPr)^1/2/(RePr ^1/2 forPr 1. Furthermore, new coordinates and dependent variables are properly defined in terms of, so that the transformed nonsimilar boundary-layer equations give numerical solutions that are uniformly valid over the entire range of mixed convection intensity from forced convection limit to free convection limit for fluids of any Prandtl number from 0.001 to 10,000. The effects of mixed convection intensity and the Prandtl number on the velocity profiles, the temperature profiles, the wall friction, and the heat transfer rate are illustrated for both cases of buoyancy assisting and opposing flow conditions.

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Mischkonvektion an einer vertikalen Platte für Fluide beliebiger Prandtl-Zahl

Zusammenfassung Für die kombinierte Zwangs- und freie Konvektion an einer isothermen senkrechten Platte wird ein Mischkonvektions-Parameter=( Ra) ^1/4 (Re)^1/2, mit=Pr/(1 +Pr) und=Pr/(1 +Pr)^1/2 vorgeschlagen, den die gebräuchliche Richardson-Zahl, Gr/Re², ersetzen soll. Dieser Parameter kann ohne weiteres auf die maßgebenden Kennzahlen für den relativen Einfluß der erzwungenen und der freien Konvektion reduziert werden,Ra ^1/4/(Re ^1/2 Pr ^1/3) fürPr 1 und (RaPr)^1/4/(RePr)^1/2 fürPr 1. Weiterhin werden neue Koordinaten und abhängige Variablen als Funktion von definiert, so daß für die transformierten Grenzschichtgleichungen numerische Lösungen erstellt werden können, die über den gesamten Bereich der Mischkonvektion, von der freien Konvektion bis zur Zwangskonvektion, für Fluide jeglicher Prandtl-Zahl von 0.001 bis 10.000 gleichmäßig gültig sind. Der Einfluß der Intensität der Mischkonvektion und der Prandtl-Zahl auf die Geschwindigkeitsprofile, die Temperaturprofile, die Wandreibung und den Wärmeübergangskoeffizienten werden für die beiden Fälle der Strömung in und entgegengesetzt zur Schwerkraftrichtung dargestellt.

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Nomenclature C _f local friction coefficient - C _p specific heat capacity - f reduced stream function - g gravitational acceleration - Gr local Grashoff number,g T _w –T )x³/v² - Nu local Nusselt number - Pr Prandtl number,v/ - Ra local Rayleigh number,g T _w –T x ³/( v) - Re local Reynolds number,u x/v - Ri Richardson number,Gr/Re ² - T fluid temperature - T _w wall temperature - T free stream temperature - u velocity component in thex direction - u free stream velocity - v velocity component in they direction - x vertical coordinate measuring from the leading edge - y horizontal coordinate Greek symbols thermal diffusivity - thermal expansion coefficient - mixed convection parameter (Ra)1/4/Re)^1/2 - pseudo-similarity variable,(y/x) - ₀ conventional similarity variable,(y/x)Re ^1/2 - dimensionless temperature, (T–T T _W –T - unified mixed-flow parameter, [(Re) ^1/2 + (Ra)^1/4] - dynamic viscosity - kinematic viscosity - stretched streamwise coordinate or mixed convection parameter, [1 + (Re)^1/2/(Ra) ^1/4]^–1=/(1 +) - density - Pr/(1 + Pr) _w wall shear stress - stream function - Pr/(l+Pr)^1/3 This research was supported by a grand from the National Science Council of ROC     

Pressure pulsations during interaction between a subsonic jet and the subsonic section of a supersonic jeton a flat obstacle

Pressure pulsations were measured during in-leakage of a subsonic jet and the subsonic section of a heated supersonic jet on a flat obstacle. Data have been obtained on the total and spectrum levels of the pressure pulsations at different spacings X of the obstacle from the nozzle exit. It is shown that when the obstacle is disposed at the section of the jet where the local velocity is subsonic, the pulsation levels outside the dependence on the conditions at the nozzle exit (Mach number Maxa 0 a 3.0; stagnation temperature T₀=280–1200K) vary in direct proportion to the local velocity head q. The ratio between the total level and q is (/g)=0.2–0.3. It is established that for a subsonic velocity ahead of the obstacle, all the spectra obtained for different values of M _a , T₀, d _a and X in the coordinates Sh=f(d/V) and (₁*/q)(V/d) will lie on a single generalized spectrum. Here ₁* is the pulsation level in a 1-Hz band, and d and V are, respectively, the jet diameter and velocity directly in front of the obstacle.Translated from Izvestiya Akademiya Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 172–174, September–October, 1975.