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All sourcesNASA NTRSDOE OSTI
NASA NTRS1951

Resistance of a plate in parallel flow at low Reynolds numbers

The present paper gives the results of measurements of the resistance of a plate placed parallel to the flow in the range of Reynolds numbers from 10 to 2300; in this range the resistance deviates from the formula of Blasius. The lower limit of validity of the Blasius formula is determined and also the increase in resistance at the edges parallel to the flow in the case of a plate of finite width.

BOUNDARY LAYER - PLATES, FLAT - LAMINAR↗
NASA NTRS1993

Boundary layer flow of air over water on a flat plate

A non-similar boundary layer theory for air blowing over a water layer on a flat plate is formulated and studied as a two-fluid problem in which the position of the interface is unknown. The problem is considered at large Reynolds number (based on x), away from the leading edge. A simple non-similar analytic solution of the problem is derived for which the interface height is proportional to x(sub 1/4) and the water and air flow satisfy the Blasius boundary layer equations, with a linear profile in the water and a Blasius profile in the air. Numerical studies of the initial value problem suggests that this asymptotic, non-similar air-water boundary layer solution is a global attractor for all initial conditions.

Nelson, John↗
NASA NTRS1994

Interaction of Oblique Instability Waves with Weak Streamwise Vortices

This paper is concerned with the effect of a weak spanwise-variable mean-flow distortion on the growth of oblique instability waves in a Blasius boundary layer. The streamwise component of the distortion velocity initially grows linearly with increasing streamwise distance, reaches a maximum, and eventually decays through the action of viscosity. This decay occurs slowly and allows the distortion to destabilize the Blasius flow over a relatively large streamwise region. It is shown that even relatively weak distortions can cause certain oblique Rayleigh instability waves to grow much faster than the usual two-dimensional Tollmien-Schlichting waves that would be the dominant instability modes in the absence of the distortion. The oblique instability waves can then become large enough to interact nonlinearly within a common critical layer. It is shown that the resulting nonlinearity is weak and that the common amplitude of the interacting oblique waves is governed by the amplitude evolution equation derived in Goldstein & Choi (1989). The implications of these results for Klebanoff-type transition are discussed.

Goldstein, M. E.↗
NASA NTRS1995

Development of a two-equation turbulence model for hypersonic flows: Evaluation of a low Reynolds number correction to the Kappa - epsilon two equation compressible turbulence model - Volume 1

The objective of the current research is the development of an improved k-epsilon two-equation compressible turbulence model for turbulent boundary layer flows experiencing strong viscous-inviscid interactions. The development of an improved model is important in the design of hypersonic vehicles such as the National Aerospace Plane (NASP) and the High Speed Civil Transport (HSCT). Improvements have been made to the low Reynolds number functions in the eddy viscosity and dissipation of solenoidal dissipation of the k-epsilon turbulence mode. These corrections offer easily applicable modifications that may be utilized for more complex geometries. The low Reynolds number corrections are functions of the turbulent Reynolds number and are therefore independent of the coordinate system. The proposed model offers advantages over some current models which are based upon the physical distance from the wall, that modify the constants of the standard model, or that make more corrections than are necessary to the governing equations. The code has been developed to solve the Favre averaged, boundary layer equations for mass, momentum, energy, turbulence kinetic energy, and dissipation of solenoidal dissipation using Keller's box scheme and the Newton spatial marching method. The code has been validated by removing the turbulent terms and comparing the solution with the Blasius solution, and by comparing the turbulent solution with an existing k-epsilon model code using wall function boundary conditions. Excellent agreement is seen between the computed solution and the Blasius solution, and between the two codes. The model has been tested for both subsonic and supersonic flat-plate turbulent boundary layer flow by comparing the computed skin friction with the Van Driest II theory and the experimental data of Weighardt; by comparing the transformed velocity profile with the data of Weighardt, and the Law of the Wall and the Law of the Wake; and by comparing the computed results of an adverse pressure gradient with the experimental data of Fernando and Smits. Good agreement is obtained with the experimental correlations for all flow conditions.

Knight, Doyle D.↗
NASA NTRS1996

Optimum Suction Distribution for Transition Control

The optimum suction distribution which gives the longest laminar region for a given total suction is computed. The goal here is to provide the designer with a method to find the best suction distribution subject to some overall constraint applied to the suction. We formulate the problem using the Lagrangian multiplier method with constraints. The resulting non-linear system of equations is solved using the Newton-Raphson technique. The computations are performed for a Blasius boundary layer on a flat-plate and crossflow cases. For the Blasius boundary layer, the optimum suction distribution peaks upstream of the maximum growth rate region and remains flat in the middle before it decreases to zero at the end of the transition point. For the stationary and travelling crossflow instability, the optimum suction peaks upstream of the maximum growth rate region and decreases gradually to zero.

Balakumar, P.↗
NASA NTRS2001

Spatial Linear Instability of Confluent Wake/Boundary Layers

The spatial linear instability of incompressible confluent wake/boundary layers is analyzed. The flow model adopted is a superposition of the Blasius boundary layer and a wake located above the boundary layer. The Orr-Sommerfeld equation is solved using a global numerical method for the resulting eigenvalue problem. The numerical procedure is validated by comparing the present solutions for the instability of the Blasius boundary layer and for the instability of a wake with published results. For the confluent wake/boundary layers, modes associated with the boundary layer and the wake, respectively, are identified. The boundary layer mode is found amplified as the wake approaches the wall. On the other hand, the modes associated with the wake, including a symmetric mode and an antisymmetric mode, are stabilized by the reduced distance between the wall and the wake. An unstable mode switching at low frequency is observed where the antisymmetric mode becomes more unstable than the symmetric mode when the wake velocity defect is high.

Liou, William W.↗
NASA NTRS1949

Report on Investigation of Developed Turbulence

The recent experiments by Jakob and Erk, on the resistance of flowing water in smooth pipes, which are in good agreement with earlier measurements by Stenton and Pannell, have caused me to change my opinion that the empirical Blasius law (resistance proportional to the 7/4 power of the mean velocity) was applicable up to arbitrarily high Reynolds numbers. According to the new tests the exponent approaches 2 with increasing Reynolds number, where it remains an open question whether or not a specific finite limiting value of the resistance factor lambda is obtained at R = infinity. With the collapse of Blasius' law the requirements which produced the relation that the velocity in the proximity of the wall varied in proportion to the 7th root of the wall distance must also become void. However, it is found that the fundamental assumption that led to this relationship can be generalized so as to furnish a velocity distribution for any empirical resistance law. These fundamental assumptions can be so expressed that for the law of velocity distribution in proximity of the wall as well as for that of friction at the wall, a form can be found in which the pipe diameter no longer occurs, or in other words, that the processes in proximity of a wall are not dependent upon the distance of the opposite wall.

Prandtl, L.↗
DOE OSTI2022

On the enhancement of boundary layer skin friction by turbulence: an angular momentum approach

Turbulence enhances the wall shear stress in boundary layers, significantly increasing the drag on streamlined bodies. Other flow features such as free stream pressure gradients and streamwise boundary layer growth also strongly influence the local skin friction. In this paper, an angular momentum integral (AMI) equation is introduced to quantify these effects by representing them as torques that alter the shape of the mean velocity profile. This approach uniquely isolates the skin friction of a Blasius boundary layer in a single term that depends only on the Reynolds number most relevant to the flow's engineering context, so that other torques are interpreted as augmentations relative to the laminar case having the same Reynolds number. The AMI equation for external flows shares this key property with the so-called FIK relation for internal flows (Fukagata et al. , Phys. Fluids , vol. 14, 2002, pp. L73–L76). Without a geometrically imposed boundary layer thickness, the length scale in the Reynolds number for the AMI equation may be chosen freely. After a brief demonstration using Falkner–Skan boundary layers, the AMI equation is applied as a diagnostic tool on four transitional and turbulent boundary layer direct numerical simulation datasets. Regions of negative wall-normal velocity are shown to play a key role in limiting the peak skin friction during the late stages of transition, and the relative strengths of terms in the AMI equation become independent of the transition mechanism a very short distance into the fully turbulent regime. The AMI equation establishes an intuitive, extensible framework for interpreting the impact of turbulence and flow control strategies on boundary layer skin friction.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1968

Internal low Reynolds number turbulent heat transfer

The results of a semi-analytical and experimental study of adiabatic tube flow and an analytical and experimental investigation of the thermal entrance region for gas flowing through electrically heated circular tubes are presented. Emphasis is placed on the low Reynolds number turbulent flow regime--defined as fully turbulent flow at bulk Reynolds numbers from 3,000 to about 15,000. Adiabatic air velocity and friction data and localized heat transfer measurements for air and helium, at low heating rates, are presented for this range. The adiabatic data were obtained in a 1.61 inch 1D tube for flow at bulk Reynolds numbers from 3,000 to 15,000. A continuous, Reynolds number-dependent, profile is developed from the data by using a modification of Reichardts' wall and middle law eddy diffusivity expressions. The velocity profile satisfies continuity. It is valid for all Reynolds numbers in excess of 3,000 for which the flow is fully turbulent and the Blasius friction factor expression is valid. The thermal entrance problem for a fully developed velocity profile is solved analytically by the method of Sparrow, Hallman, and Siegel. The solution is based on the profile developed from the velocity study. Tabular values of the eigenvalues and normalized Nusselt numbers for gases are presented for it range of Reynolds numbers from 3,000 to 50,000. The axial variation of Nusselt number is found to be correlated by [Nu/Nu_(∞)] = 1 + 0.8 (1+ 70,000 Re^(-3/2)) ((x/D)^(-1)) to within ± 5 per cent for x/D ≥2. The fully developed value agrees with the Dittus - Boelter correlation, Nu_(∞) = 0.021 Re^(0.8) Pr^(0.4) For the eigenvalues, λ^(2)_(n), and the associated constants, A_(n), correlations of the form λ^(2)_(n) =A_(1,n) Re^(-b_(1,n) + C_(1,n) Re^(-d_(1,n)) A_(n) = -A_(2,n) Re^(-b_(2,n) + C_(2,n) Re^(-d_(2,n)) are obtained. The coefficients and powers are presented in tabular form. Heat transfer data are presented, primarily for helium, for the low Reynolds number turbulent range. A one-quarter inch, resistively heated, vertical, circular tube was used for the study. The data cover an axial range from 1.2 to 96 diameters; wall-to-bulk temperature ratios vary from 1 to 1.4. In the low Reynolds number turbulent regime, these data clearly support the present analytical solution rather than the prediction obtained by applying the eddy diffusivity distribution used by Sparrow, Hallman, and Siegel.

Harold C. Reynolds, Jr.↗
NASA NTRS1971

Finite difference solution to the three-dimensional, incompressible thermal energy boundary-layer equation

An implicit numerical method has been adopted for the solution to the three-dimensional, energy boundary-layer equation. The energy equation is written in terms of a dimensionless temperature function, the relative stagnation-enthalpy difference, and transformed by the introduction of a Blasius-type transformation of coordinates as well as dimensionless stream functions. The method is applied to the problem consisting of an infinite cylinder joined with its axis perpendicular to a thin, flat, heated plate. A Prandtl number equal to one is simply considered.

Fillo, J. A.↗
NASA NTRS1972

Computer program for calculating laminar and turbulent boundary layer development in compressible flow

A computer program is described which performs a numerical integration of the equations of motion for a compressible two-dimensional boundary layer. Boundary layer calculations may be carried out for both laminar and turbulent flow for arbitrary Reynolds number and free stream Mach number distribution on planar or axisymmetric bodies with wall heating or cooling, longitudinal wall curvature, wall suction or blowing, and a rough or a smooth wall. A variety of options are available as initial conditions. The program can generate laminar initial conditions such as Falkner-Skan similarity solutions (so that initial wedge flows can be simulated including Blasius or stagnation point flow) or approximate equilibrium turbulent profiles. Alternatively, initial profile input data can be utilized.

Herring, H. J.↗
NASA NTRS1973

A study of the effect of a boundary layer profile on the dynamic response and acoustic radiation of flat panels

The response of a thin, elastic plate to a harmonic force which drives the plate from below and a compressible air stream with a viscous boundary layer flowing parallel to the upper surface along the length was investigated. Equations governing the forced response of the coupled plate-aerodynamic system are derived along with appropriate boundary conditions. Calculations of basic solution parameters for a linear velocity profile and for a Blasius profile showed that the same system response could be obtained from each profile if appropriate values of boundary layer thickness were chosen for each profile.

Mixson, J. S.↗
NASA NTRS1952

Experimental Investigation of Average Heat-Transfer and Friction Coefficients for Air Flowing in Circular Tubes Having Square-Thread-Type Roughness

An investigation of forced-convection heat transfer and associated pressure drops was conducted with air flowing through electrically heated Inconel tubes having various degrees of square-thread-type roughness, an inside diameter of 1/2 inch, and a length of 24 inches. were obtained for tubes having conventional roughness ratios (height of thread/radius of tube) of 0 (smooth tube), 0.016, 0.025, and 0.037 over ranges of bulk Reynolds numbers up to 350,000, average inside-tube-wall temperatures up to 1950deg R, and heat-flux densities up to 115,000 Btu per hour per square foot. Data The experimental data showed that both heat transfer and friction increased with increase in surface roughness, becoming more pronounced with increase in Reynolds number; for a given roughness, both heat transfer and friction were also influenced by the tube wall-to-bulk temperature ratio. Good correlation of the heat-transfer data for all the tubes investigated was obtained by use of a modification of the conventional Nusselt correlation parameters wherein the mass velocity in the Reynolds number was replaced by the product of air density evaluated at the average film temperature and the so-called friction velocity; in addition, the physical properties of air were evaluated at the average film temperature. The isothermal friction data for the rough tubes, when plotted in the conventional manner, resulted in curves similar to those obtained by other investigators; that is, the curve for a given roughness breaks away from the Blasius line (representing turbulent flow in smooth tubes) at some value of Reynolds number, which decreases with increase in surface roughness, and then becomes a horizontal line (friction coefficient independent of Reynolds number). A comparison of the friction data for the rough tubes used herein indicated that the conventional roughness ratio is not an adequate measure of relative roughness for tubes having a square-thread-type element. The present data, as well as those of other investigators, were used to isolate the influence of ratios of thread height to width, thread spacing to width, and the conventional roughness ratio on the friction coefficient. A fair correlation of the friction data was obtained for each tube with heat addition when the friction coefficient and Reynolds number were defined on the basis of film properties; however, the data for each tube retained the curve characteristic of that particular roughness. The friction data for all the rough tubes could be represented by a single line for the complete turbulence region by incorporating a roughness parameter in the film correlation. No correlation was obtained for the region of incomplete turbulence.

Sams, E. W.↗
NASA NTRS1975

Numerical calculation of finite amplitude effects in unstable laminar boundary layers

A numerical study of one stage of the transition region from laminar to turbulent flow of the boundary layer over a flat plate is performed. Benney-Lin theory is applied to the laminar profile, which is closer to that of the experiment, i.e., the Blasius profile. Several laminar profiles are studied, all belonging to the Falkner-Skan family. The flow is found to depend principally upon the disturbance spanwise wavenumber, beta, whose variation results, in some instances, in a system of counter-rotating vortices, one on top of the other, for the mean secondary flow.

Antar, B. N.↗
NASA NTRS1975

Use of numerically generated body-fitted coordinate systems for solution of the Navier-Stokes equations

A procedure for numerical solution of the time-dependent, two-dimensional incompressible Navier-Stokes equations that can treat the unsteady laminar flow about bodies of arbitrary shape, such as two-dimensional airfoils, multiple airfoils, and submerged hydrofoils, as naturally as it can deal with the flow about simple bodies. The solution is based on a method of automatic numerical generation of a general curvilinear coordinate system with coordinate lines coincident with all boundaries of a general multiconnected region containing any number of arbitrarily shaped bodies. The curvilinear coordinates are generated as the solution of two elliptical partial differential equations with Dirichlet boundary conditions, one coordinate being specified to be constant on each of the boundaries, and a distribution of the other being specified along the boundaries. The solution compares excellently with the Blasius boundary layer solution for the flow past a semiinfinite flat plate.

Thompson, J. F.↗
NASA NTRS1975

A numerical method for the prediction of high-speed boundary-layer transition using linear theory

A method is described of estimating the location of transition in an arbitrary laminar boundary layer on the basis of linear stability theory. After an examination of experimental evidence for the relation between linear stability theory and transition, a discussion is given of the three essential elements of a transition calculation: (1) the interaction of the external disturbances with the boundary layer; (2) the growth of the disturbances in the boundary layer; and (3) a transition criterion. The computer program which carried out these three calculations is described. The program is first tested by calculating the effect of free-stream turbulence on the transition of the Blasius boundary layer, and is then applied to the problem of transition in a supersonic wind tunnel. The effects of unit Reynolds number and Mach number on the transition of an insulated flat-plate boundary layer are calculated on the basis of experimental data on the intensity and spectrum of free-stream disturbances. Reasonable agreement with experiment is obtained in the Mach number range from 2 to 4.5.

Mack, L. M.↗
NASA NTRS1975

Numerical solutions of the unsteady Navier-Stokes equations for arbitrary bodies using boundary-fitted curvilinear coordinates

A method of automatic body-fitted curvilinear coordinate generation is described and used to construct a finite-difference solution of the full incompressible time-dependent Navier-Stokes equations for the unsteady laminar viscous flow arbitrary two-dimensional airfoils or any other two-dimensional body. A method of controlling the spacing of the coordinate lines encircling the body is developed in order to treat high Reynolds number flows, since the coordinate lines must concentrate near the surface to a greater degree as the Reynolds number increases. Multiple airfoils and submerged hydrofoils are treated as illustrative examples. The solution shows good agreement with the Blasius boundary layer solution for the flow past a semi-infinite flat plate.

Thompson, J. F.↗