Karman Constant Research Articles

Lag model for turbulent boundary layers over rough bleed surfaces

Boundary-layer mass removal (bleed) through spanwise bands of holes on a surface is used to prevent or control separation and to stabilize the normal shock in supersonic inlets. The addition of a transport equation lag relationship for eddy viscosity to the rough wall algebraic turbulence model of Cebeci and Chang was found to improve agreement between predicted and measured mean velocity distributions downstream of a bleed band. The model was demonstrated for a range of bleed configurations, bleed rates, and local freestream Mach numbers. In addition, the model was applied to the boundary-layer development over acoustic lining materials for the inlets and nozzles of commercial aircraft. The model was found to yield accurate results for integral boundary-layer properties unless there was a strong adverse pressure gradient. Nomenclature A = constant (26.0) in van Driest's damping function, Eq. (3) ks = equivalent sand-grain roughness height k+ = roughness Reynolds number, ksuT/v k^ = constant (0.02) in outer layer eddy viscosity model, Eq. (4) k3 = constant (0.5) in the lag model, Eq. (7) / = length scale rabl = bleed mass flow rate p = static pressure R = augmented mixing length due to surface roughness effect Re = Reynolds number M, v = velocity components in the streamwise and the normal to surface directions UT = frictional velocity *, y = coordinates in the streamwise and the normal to boundary-layer surface directions T = intermittency factor in Eq. (4), 0 < F < 1 8* = displacement thickness 6 = momentum thickness K = von Karman constant, 0.40, Eq. (3) fji — viscosity p = density v = kinematic viscosity of the fluid Subscripts e = outer region of boundary layer eq = equilibrium turbulent flow t = turbulent flow Superscripts ' = fluctuating turbulent quantity + = quantity normalized by vluT

Turbulence models for compressible boundary layers

It is shown that to satisfy the general accepted compressible law of the wall derived from the Van Driest transformation, turbulence modeling coefficients must actually be functions of density gradients. The transformed velocity profiles obtained by using standard turbulence model constants have too small a value of the effective von Karman constant kappa in the log-law region (inner layer). Thus, if the model is otherwise accurate, the wake component is overpredicted and the predicted skin friction is lower than the expected value.

Strain‐Induced Interaction of Substitutional Atoms in Cubic Metals

AbstractThe energies of strain‐induced (elastic) interaction of pairs of substitutional atoms and host atom displacements around a substitutional atom are calculated for Al, Ag, Au, Cu, Ni, Pd, Cr, αFe, Mo, Nb, Ta, V, and W taking into account the discrete atomic structure of the host lattice. The elastic constants, Born‐von Karman constants of the host lattice, and coefficients of the concentration expansion of the solid solution lattice due to substitutional atoms are the numerical parameters used. A computer calculation is carried out in the general form appropriate for any substitutional atoms in these metals. The energies of strain‐induced interaction of substitutional atoms are of the same order as the “chemical” interaction energy and should be taken into account during computer simulation of structure and properties of the substitutional solid solutions.

Evaluation of von Karman's Constant from Integral Flow Parameters

Theory and data for turbulent flow in smooth pipes are employed to evaluate von Karman's constant from integral flow parameters. A set of equations for mean velocity and centerline velocity are der...

Experimental study in a rotating channel on the similarity law of tracer concentration distribution in the turbulent Ekman boundary layer

Concentration disrribution of airborne pollutants released into the atmosphere by large and elevated industrial sources or extended urban areas generally depends on the dynamical and thermal structure of the whole depth of the atmospheric planetary boundary layer (PBL). In order to be able to get meaningful predictions of pollutant dispersion from simple dispersion formulae using surface parameters as input data (like the Gaussian one), it is necessary to establish some reliable relationships between these surface parameters and the whole PBL's vertical structure. This paper presents the results of laboratory experiments in rotating hydraulic channel, aimed at investigating the influence of the Earth rotation on the dispersion of tracers which spread in a near-neutral Ekman turbulent PBL and at expressing the change of u and v components of mean wind with height in terms of rotation and turbulence scales typical of the atmospheric PBL. The results provide some evidence that in a near-neutral PBL the profiles of dilution ratio, adimensionalized by means of friction velocity u ∗ = √ u′w′ and Coriolis parameter f = 2Ω sinϕ , as a function of an adimensional height zf χ u ∗ , where χ is the von Karman constant, can be described by a universal function for a given Rossby number of the flow. This result confirms the theoretical expectation put forth by Wippermann and Yordanov in their study on the prediction of concentration patterns in a rotating turbulent PBL.

Modeling supersonic inlet boundary-layer bleed roughness

Boundary-layer mass removal (bleed) through spanwise bands of holes on a surface is used to prevent or control separation in supersonic inlets. The rough wall algebraic turbulence model of Cebeci and Chang was added to both boundary-layer and Navier-Stokes analyses to simulate the overall effect of bleed on the growth of a boundary layer. Roughness values were determined for seven bleed configurations, a range of Mach numbers between 1.3-4, and bleed rates between zero and choked values. For the bleed experiments considered, the roughness was found to be a function of the fraction of the upstream boundary-layer mass flux removed. Choked bleed flow through holes at a low angle, with respect to the surface, minimized the roughness effect and gave the best improvement in the boundary-layer velocity distribution for separation control. Nomenclature A+ = Van Driest parameter, 26 d ~ bleed hole diameter k = von Karman constant, 0.4 ks = equivalent sand grain roughness Lid — hole aspect ratio / = local turbulent length scale M = Mach number N = number of rows of bleed holes in bleed band P/POO = local static pressure/freestream static pressure R = roughness parameter, in. ulue — ratio of local to freestream velocity within the boundary layer «r = (TJPJ1 u' = local turbulent velocity scale

A comparison of airborne eddy correlation and bulk aerodynamic methods for ocean-air turbulent fluxes during cold-air outbreaks

Four bulk schemes (LKB, FG, D and DB), with the flux-profile relationships of Liuet al. (1979), Francey and Garratt (1981), Dyer (1974), and Dyer and Bradley (1982), are derived from the viscous interfacial-sublayer model of Liuet al. These schemes, with stability-dependent transfer coefficients, are then tested against the eddy-correlation fluxes measured at the 50 m flight level above the western Atlantic Ocean during cold-air outbreaks. The bulk fluxes of momentum (τ), sensible heat (H), and latent heat (E) are found to increase with various von Karman constants (k M for τk H forH, andk E forE). Except that the LKB scheme overestimates by 28% (46Wm−2), on the average, the fluxes estimated by the four bulk schemes appear to be in fairly good agreement with those of the eddy correlation method (magnitudes of biases within 10% for τ, 17% forH, and 13% forE). The results suggest that the overall fluxes and surface-layer scaling parameters are best estimated by FG and thatk H <k E . On the average, the FG scheme underestimates τ by 10% (0.032N m−2) andE by 4% (12Wm−2), and overestimatesH by 0.3% (0.5W m−2). The equivalent neutral transfer coefficients at 10 m height of the FG scheme compare well with some schemes of those tested by Blanc (1985).

The implications of linearly varying eddy viscosity for wind-driven current profiles

A major problem in numerical circulation modelling is the specification of vertical turbulence closure. After the simplicity of a constant eddy viscosity, the next step towards reality appears to be an eddy viscosity described by υ = κ u * z where κ is von Karman's constant, u * is a friction velocity, and z is the distance away from the sea surface or bottom. Applied to the “Ekman problem”, this eddy viscosity results in current profiles that have logarithmic boundary layers. The Ekman depth is replaced by a new depth of influence, κu */ f (where f is the Coriolis frequency). If this depth is less than the water depth, the wind-forced current profile does not reach the bottom, and is unaffected by the bottom conditions. For water depths less than the depth of surface influence, currents through the whole water column are sensitive to the value of the bottom roughness. By contrast, changes to the surface roughness only affect currents in a thin layer near the surface. Numerically, resolution of the log-layers at the sea surface and bottom requires the use of special techniques. The log-layers may be described analytically, or the vertical coordinate may be transformed to expand the near-boundary regions. For modelling the mid-water column, away from the log-layers, a constant eddy viscosity with a linear bottom slip condition represents the currents with reasonable accuracy. Appropriate values of the constant eddy viscosity and bottom friction parameter are determined empirically from the shear stress and bottom roughness by formulae that are consistent over a broad parameter range.

Dilatation-dissipation corrections for advanced turbulence models

This paper analyzes dilatation-dissipation based compressibility corrections for advanced turbulence models. Numerical computations verify that the dilatation-dissipation corrections devised by Sarkar and Zeman greatly improve both the k-omega and k-epsilon model predicted effect of Mach number on spreading rate. However, computations with the k-gamma model also show that the Sarkar/Zeman terms cause an undesired reduction in skin friction for the compressible flat-plate boundary layer. A perturbation solution for the compressible wall layer shows that the Sarkar and Zeman terms reduce the effective von Karman constant in the law of the wall. This is the source of the inaccurate k-gamma model skin-friction predictions for the flat-plate boundary layer. The perturbation solution also shows that the k-epsilon model has an inherent flaw for compressible boundary layers that is not compensated for by the dilatation-dissipation corrections. A compressibility modification for k-gamma and k-epsilon models is proposed that is similar to those of Sarkar and Zeman. The new compressibility term permits accurate predictions for the compressible mixing layer, flat-plate boundary layer, and a shock separated flow with the same values for all closure coefficients.

The turbulent kinetic energy budget in the atmospheric surface layer: A review and an experimental reexamination in the field

The one-dimensional equation for the turbulent kinetic energy budget in steady, horizontally-homogeneous flow near the ground is reviewed, and some of the many experimental evaluations of its stability-dependent terms obtained during the last twenty years are compared. Uncertainties attributable to instrument error and inadequate sites are discussed, and it is demonstrated that improved equipment makes it possible to evaluate contributions to the budget with comparatively simple experiments. A preliminary field study finds a von Karman constant of k=0.387±0.016 and a wind shear function for the unstable surface layer% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdy2aaS% baaSqaaiaad2gaaeqaaOGaeyypa0JaaiikaiaaigdacqGHsislcaaI% YaGaaGOmaiaac6cacaaI2aGaamOEaiaac+cacaWGmbGaaiykamaaCa% aaleqabaGaeyOeI0IaaGymaiaac+cacaaI0aaaaaaa!4587!\[\phi _m = (1 - 22.6z/L)^{ - 1/4} \]: the form % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaig% dacqGHsislcaaIXaGaaGynaiaac6cacaaIXaGaamOEaiaac+cacaWG% mbGaaiykamaaCaaaleqabaGaeyOeI0IaaGymaiaac+cacaaIZaaaaa% aa!419C!\[(1 - 15.1z/L)^{ - 1/3} \] fits equally well over the limited range of instability observed. Turbulence dissipation is found to be 15 to 20% too small to balance the production of energy by wind shear in the neutral surface layer, and this deficit appears to remain approximately constant relative to the total rate of energy production as instability increases to% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEaiaac+% cacaWGmbGaeyypa0JaeyOeI0IaaGimaiaac6cacaaIXaGaaGOmaaaa% !3D45!\[z/L = - 0.12\]. Renormalized dissipation rates originally measured by others are shown to exhibit similar behavior beyond this narrow range. Combining these results with those of the present study suggests a dissipation function of the form % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdy2aaS% baaSqaaiabew7aLbqabaGccqGH9aqpcqaHgpGzdaWgaaWcbaGaamyB% aaqabaGccqGHsislcaWG6bGaai4laiaadYeacqGHRaWkcaWGJbaaaa!42A3!\[\phi _\varepsilon = \phi _m - z/L + c\] in which c = -0.16 represents a near constant, net negative contribution made by the sum of the divergent transport terms.

Methods of measuring wind stress over a water surface ? Discussions of displacement height and von Karman constant

Experiments were conducted in a wind-wave tank to study the long-standing discrepancy between wind stresses over a water surface determined with the profile and eddy-correlation methods. Inasmuch as the eddy-correlation method measures the turbulence flux directly, such a discrepancy is considered as an ‘error’ in the way the profile data are analyzed. Uncertainties in selections of the zero reference plane for the logarithmic velocity profile and of the value for the von Karman constant are suggested to be major sources of the error. To match the results obtained with two methods, the displacement height needs to vary with wave conditions, and the value of the von Karman constant is also discussed.

Differing development of the velocity profiles of three-dimensional turbulent boundary layers

i x,z d •• Reynolds number based on boundarylayer thickness, = U^d/v • main-flow component of mean velocity in x direction : friction velocity, = VTV//O = freestream velocity at the edge of the boundary layer = scaling velocities of the wake function, Eqs. (7) and (12) = differences of the scaling velocities of the wake function, Eqs. (7) and (12) = nondimensional velocity component in x direction, = U/UTX • crossflow component of mean velocity in z direction = nondimensional velocity component in z direction, = W/UTZ nondimensional, characteristic quantity, = RedwK = y + K § = local coordinate aligned with the free streamline = local coordinate normal to the wall = nondimensional wall distance, = yuT/v local coordinate normal to the free streamline = angle of the free streamline = cross flow angle = angle of the wall shear stress = boundary-layer thickness = nondimensional wall distance, =y/d -von Karman constant, =0.41 = kinematic viscosity = fluid density = wall shear stress = wake function, Eq. (9) = model function, Eq. (10) = nondimensional friction velocity, = UT/

On the sensitivity of mesoscale models to surface-layer parameterization constants

The Colorado State University standard mesoscale model is used to evaluate the sensitivity of one-dimensional (1D) and two-dimensional (2D) fields to differences in surface-layer parameterization “constants”. Such differences reflect the range in the published values of the von Karman constant, Monin-Obukhov stability functions and the temperature roughness length at the surface. The sensitivity of 1D boundary-layer structure, and 2D sea-breeze intensity, is generally less than that found in published comparisons related to turbulence closure schemes generally.

Finite element analysis of the two-dimensional asymmetric relaxing channel flow by the k-.EPSILON. model.

A high-Reynolds-number k-ε model is used to predict the relaxation process which develops in the channel whose upstream half wall is made rough. Experimental evidence shows that after the step change in wall roughness, the mixing length increases considerable over the value expected in the ordinary boundary layer. It is also shown that the convection term in the turbulence kinetic energy equation is not negligible and the existence of the wall equilibrium layer is not expected. Therefore, two new refinements are applied to the boundary condition adjacent to the wall. One is that the Karman constant in the logarithmic law of the wall is assumed to vary with streamwise distance. The other is that all the terms in the turbulence energy equation are considered. It is found that the prediction of the variation in the mean velocity is improved and the abrupt decrease in the turbulence kinetic energy, which is found in the usual model, disapears.

An open question regarding shear flow with suspended sediments

This paper deals with the critical re-analysis of a series of experimental laboratory data (Coleman, 1981), concerning velocity profiles in a free surface stream, when suspended solid transport is present. The interpretation of results presented here is rather different from Coleman's original interpretation, which states that density stratification does not affect inner layer variables, e.g. von Karman constant. This difference is important, because Coleman's data are commonly cited as a proof of the above statement. In the present work the estimate of parameters is based on well known mathematical methods. Therefore the reliability of the results should be assured. So, when suspended sediment transport is present, von Karman ≪constant≫ variability seems to be an open question.

Development of a criterion for defining spray drift

Finite rate diffusion theory is developed to allow for a sedimentation component in the diffusion equations. Theory shows that the condition in which the wave front of the upper boundary of a dispersing cloud moves horizontally is consistent with the ratio v s u ∗ = b (b ⋍ 1.3) in which v s is the settling velocity and u ∗ the friction velocity in the adiabatic surface layer. For the case v s u ∗ > b the plume is forced downwards due to the dominance of sedimentation. In this situation, the distance x max over which the airborne fraction persists is given by x max ⋍ h ln (h/(ez 0))/(k( v s u ∗ −b)) where h is the release height, z 0 the roughness length and k von Karman's constant. Theoretical predictions are shown to be broadly consistent with experimental data for droplets with diameters in the size range of 10–240 μm.

Turbulent oscillatory flow over rough beds

Velocity measurements are presented for turbulent oscillatory flow over rough beds. Two components of velocity were measured with a laser-Doppler anemometer and the rough beds consisted of a single layer of sand, gravel or pebbles on a flat surface. Turbulence intensities showed significant variation during the course of the cycle. Maximum turbulence intensity propagated out from the bed at a more or less constant velocity for all beds. Variation of time-mean turbulence intensity with height was qualitatively similar to that observed in steady flows. Reynolds stress showed several interesting features. Near the bed, maximum Reynolds stress was in phase with one of the two peaks of turbulence intensity but further out it was in phase with the other, i.e. the phase of maximum Reynolds stress showed a 180° phase shift at a certain height above the bed. A related effect was seen in the time-mean eddy viscosity which was negative near the bed but positive further out. It is suggested that these effects are caused by the jets of fluid associated with vortex formation and ejection in oscillatory flow over rough beds. Maximum Reynolds stress was also significantly less than the horizontal force per unit area of bed obtained from the momentum integral. Eddy viscosity and mixing length were found to vary significantly during the course of the cycle. Variation with height of time-mean values of these variables showed similar trends, except in the near-bed region, to those observed in steady flow but derived values of the Karman constant were significantly lower. Non-dimensional defect velocity appeared to show dependence on a/ks as well as on y/δ in the outer layer away from the bed, even at high Reynolds numbers.

Measurements of near‐surface shear in the ocean

In an effort to measure current shear very near the sea surface a string of vector‐measuring current meters suspended beneath a surface float was deployed 180 km off southern California at 34°N 121°30′W on November 26, 1981 and allowed to drift freely for over 6 days. This arrangement, which we call the current meter drifter (CMD), gave measurements at depths of 2.5, 5.5, 8.5, 11.5, 14.5, 25, and 63 m. During the first two days, when the winds were light (≃8 m/s) and variable in direction, a nearly uniform current shear was observed in the upper 15 m with a low‐frequency velocity difference of 5 cm/s between the instruments at 2.5 and 14.5 m. During the last four days, when the winds were brisk (&gt; 12 m/s) and steady in direction from the NNW, a strong downwind shear of order 10−2 s−1 was observed in the upper 10 m with a velocity difference of ∼7 cm/s between the instruments at 2.5 and 8.5 m. During this same period the shear below 10 m was much smaller. The average currents during the CMD drift veer to the right of the wind stress with angular displacement increasing with depth. Time series of the velocity difference between 2.5 and 5.5 m compare very well with θ(t)u*κ−1 ln (5.5/2.5), where θ(t) is the wind direction vector (of unit magnitude), u* = (wind stress/ρ)½ is the friction velocity in water, and κ = 0.4 is von Karman's constant. On the other hand, a similar comparison of the velocity difference between 5.5 and 8.5 m to θ(t)u*κ−1In (8.5/5.5) is much poorer with observed velocity difference being much larger, possibly due to stable stratification effects. Possible errors in the measurements have been considered and estimated as less than the observed velocity differences. Near‐surface shears as large as the observed are very important in closing the momentum budget for the oceanic boundary layer.

Momentum transfer to plant canopies: Influence of structure and variable drag

Theory is developed to assess momentum transfer to vegetation canopies displaying a ‘dynamic’ response to variations in the level of mean wind-speed. For a broad range of conditions it is demonstrated that αku h(h−d)/(u ∗h) = 1.11 ± 0.17 (95 % confidence interval) with: α, as the wind-speed extinction coefficient in the upper canopy; k, von Karman's constant; u h , the wind-speed at h, the canopy height; u ∗ , the friction velocity and d, the zero plane displacement. The above relationship together with background discussion lends support to the assumption that τ = K(d u ̄ /dz) can be a reasonable approximation in the upper canopy with: τ as the shear stress, K, the momentum diffusivity and (d u ̄ /dz) the velocity gradient. With the assumption of an exponential variation of wind-speed with height in the upper layers of a plant canopy and the use of a wind-speed dependent drag coefficient C d , theory is developed to describe the profiles of shear-stress and momentum diffusivity within canopies of contrasting vertical structure. In a uniform canopy it is shown that the wind-speed and diffusivity profiles are dissimilar except for the case of C d independent of wind-speed. For canopies with a vertical structure of Gaussian form, the momentum diffusivity in the upper layers may exceed the interface value, but decreases in lower layers. From analysis of wind profiles measured within and above a cotton canopy, it is shown that the effective drag coefficient of foliage elements increases with wind-speed. The diffusivity profiles within the canopy depend to first order upon the profiles of effective foliage area participating in momentum exchange. There are contrasting views as to how this may be determined for the actual foliage area profile.

Profile of Suspended Sediment Due to Wave Action

A knowledge of sediment transport due to wave action is required to understand the dynamic process of nearshore morphology; however, these phenomena of sediment transport due to waves are complicated by the unsteady characteristics of the flow, the presence of turbulence and the interactions between the fluids and the sediment. These unsteady phenomena are much more complicated than those of steady uniform flows. Profiles of mean and unsteady concentrations are derived from the mass conservation equation using the diffusion coefficient profile adapted from the eddy viscosity profile, taking into account changes of the friction factor and boundary layer thickness in the presence of suspension. The theoretical profiles are expressed explicitly as functions of the friction factor, velocity and shear stress profile parameters, and the settling velocity of the sediment relative to the fluid velocity. The tabulated data available and the present experimental data of the mean concentration are used to fit the theoretical profile. The analysis shows that the friction factors in the presence of suspension decrease from the sediment‐free values in the same manner and magnitude as the von Karman constant.