J. Fluids Eng. 1983;105(3):251-256. doi:10.1115/1.3240982.

Eleven near-wall similarity models for three-dimensional turbulent boundary layers which have been identified in the literature are reviewed. Each model summary includes a brief review of its derivation, discusses limitations in the derivation, estimates the applicable y+ range, and compares differences among the models. This review of three-dimensional similarity models was developed as part of a larger study which tests the validity of ten of these different models by comparison with experimental data which includes the direct and simultaneous measurement of the local wall shear stress direction and magnitude in a three-dimensional turbulent flow. A direct force measurement of local wall shear stress is necessary to test the local wall shear-shear velocity relationship, τ0 = ρq*2 , generally assumed in three-dimensional flows. This review is necessary to acquaint the reader with the similarities and differences among the models tested in companion papers since differences among some of the models are significant, particularly in the coordinate systems of the vector models.

Commentary by Dr. Valentin Fuster
J. Fluids Eng. 1983;105(3):257-262. doi:10.1115/1.3240983.

Ten of eleven three-dimensional near-wall similarity models identified in the literature are evaluated with direct wall shear, velocity field, and pressure gradient data from a three-dimensional pressure-driven boundary layer flow. In a primary focus in the interval 50<y+ <300 graphical results indicate that six simpler models and the streamwise component of one complex model are adequate for profiles with monotone increasing skew up to about 15 deg. The three remaining complex models provide a better predictive capability (for the main flow component) for monotone increasing skew up to almost 20 deg but these require significantly more input. One of three transverse models shows reasonably good predictive capability. Similar general results also appear for profiles with increasing-decreasing skew as occurs with freestream streamline recurvature with the maximum skew limited to about 10 deg. In a secondary focus in the interval of y+ <50 there is a very strong tendency for the data to follow the well accepted, two-dimensional like behavior often identified with a transition or buffer region below the two-dimensional log-like law.

Commentary by Dr. Valentin Fuster
J. Fluids Eng. 1983;105(3):263-269. doi:10.1115/1.3240984.

Ten of eleven proposed three-dimensional similarity models identified in the literature are evaluated with direct wall shear, velocity field, and pressure gradient data from a three-dimensional shear-driven boundary layer flow. Results define an upper limit on velocity vector skewing for each model’s predictive ability. When combined with earlier results for pressure-driven flows, each model’s predictive ability with and without pressure gradients is summarized.

Commentary by Dr. Valentin Fuster
J. Fluids Eng. 1983;105(3):270-276. doi:10.1115/1.3240985.

As an extension to the inviscid gas flow particle trajectory model presented in earlier papers, a complementary model has been developed to establish the effect of the blade boundary layer on the trajectories of particles and thus on the resulting erosion and/or deposition. The method consists essentially in tracing particles inside the boundary layer with initial conditions taken from the inviscid flow model. The flow data required for the particle trajectory calculations are obtained by using a compressible boundary layer flow computer program. This model has been applied to the first stage stator of a large electric utility gas turbine operating with coal gas. Results are compared with the predictions of the inviscid flow model. It is shown that the effect of the boundary layer on the trajectories of particles smaller than 6 μm is important. Since the hot gas cleaning system of a pressurized fluidized-bed gasifier system is projected to remove particles larger than 6 μm diameter effectively, it is concluded that an accurate assessment of turbine erosion and deposition requires inclusion of the boundary layer effect. Although these results emphasize the relative importance of the blade boundary layer, the absolute accuracy of the method remains to be demonstrated and is thought to be largely dependent on the basic data concerning the erosivity and sticking probability of particles.

Commentary by Dr. Valentin Fuster
J. Fluids Eng. 1983;105(3):277-283. doi:10.1115/1.3240986.

Motion of air bubbles in a high-specific-speed axial-flow pump impeller was analyzed on the basis of measured streak lines of air bubbles in the impeller. The results were compared with those obtained by a numerical solution of the bubble motion equations for three dimensional flow. Governing factors of the bubble motion are the drag force due to the surrounding water and the force due to the pressure gradient. Trajectories of the bubbles deviate somewhat from the streamlines of water, and the amount of the deviation is dependent on the bubble diameter and also on specific-speeds of the pumps and flow rate of water.

Commentary by Dr. Valentin Fuster
J. Fluids Eng. 1983;105(3):284-289. doi:10.1115/1.3240989.

Mean flow measurements, and some turbulence measurements, have been made in a two-dimensional incompressible constant-pressure (“flat plate”) turbulent boundary layer beneath a nearly homogeneous nearly-isotropic (grid-generated) turbulent free stream. An appreciably nonlinear dependence of the skin-friction coefficient and other boundary layer parameters on rms free-stream turbulence intensity has been confirmed. A much wider range of free-stream length scales has been studied than in previous work, and the results (which agree well with previous data where they overlap) clearly indicate the large effect of free-stream length scale on the response of the boundary layer. The decrease of free-stream turbulence effect with increasing length scale is at least partly attributable to simple reduction of normal-component velocity fluctuations by the solid surface; this would not be the case in free shear layers.

Commentary by Dr. Valentin Fuster
J. Fluids Eng. 1983;105(3):290-296. doi:10.1115/1.3240990.

A closed-form analytical solution is developed to hitherto unsolved problem of steady laminar flow of a Newtonian fluid in the entrance region of elliptical ducts. The analysis is based on the Karman-Pohlhausen integral method and entails solution of the integrated forms of the mass and the momentum balance equations. According to this analysis, the hydrodynamic entrance length based on 99 percent approach to the fully developed flow is equal to 0.5132λ/(1+λ2 ) where λ is the aspect ratio. Also, the fully developed incremental pressure defect is found to be 7/6 which is independent of the aspect ratio. In the limit when the flow becomes fully developed, the solution converges to the known exact asymptotic solution. Available, wide-ranging velocity measurements for a circular tube agree with the analytical predictions within 7 percent. Also, available pressure drop measurements near the inlet of a circular tube agree with the analytical predictions within 2 percent.

Commentary by Dr. Valentin Fuster
J. Fluids Eng. 1983;105(3):297-302. doi:10.1115/1.3240991.

Critical mass flux and axial pressure profile data for fluid nitrogen are presented for N = 20, 15, 10, and 7 N-sequential-orifice-inlet configurations uniformly spaced at 15.5 cm. These data correlate well over a wide range in reduced temperature (0.7 < Tr, 0 < ambient) and reduced pressure (to Pr = 2) and are in general agreement with previous studies of one to four inlets. Experimental and theoretical agreement is good for liquid and gas critical mass flux, but inconclusive in the near-thermodynamic critical regions.

Commentary by Dr. Valentin Fuster
J. Fluids Eng. 1983;105(3):303-307. doi:10.1115/1.3240992.

It is shown that an axisymmetric solution of the Navier-Stokes equations can be obtained for potential flow superimposed on Poiseuille flow. The result is used here to obtain a fully developed solution for flow in a porous pipe with variable suction or injection and to show how to obtain the suction distribution needed to change a specified axial velocity distribution at one cross-section to a specified axial velocity distribution at another cross-section.

Commentary by Dr. Valentin Fuster
J. Fluids Eng. 1983;105(3):308-318. doi:10.1115/1.3240993.

This paper establishes the drag characteristics of finite cylinders of aspect ratio 1, 4, 10 and 100 for Reynolds numbers less than 1000 including the viscous regime. The effect of the drag and vortex shedding characteristics of curving a finite cylinder into a toroidal shape is investigated. The curvature reduces drag by as much as 13 percent over its linear counterpart in the viscous regime. Vortex shedding characteristics of tori include all the features of cylinders in addition to a solidity range that behaves like solid bodies and an intermediate range where two vortex flow patterns can exist. These patterns can occur either as alternating ring vortices or a less common but more stable counterrotating helical vortex pair.

Commentary by Dr. Valentin Fuster
J. Fluids Eng. 1983;105(3):323-328. doi:10.1115/1.3240998.

The original derivation of the basic theory governing the aerodynamics of both hovercraft and modern floatation ovens, requires the validity of some extremely crude assumptions. However, the basic theory is surprisingly accurate. It is shown that this accuracy occurs because the final expression of the basic theory can be derived by approximating the full Navier-Stokes equations in a manner that clearly shows the limitations of the theory. These limitations are used in discussing the relatively small discrepancies between the theory and experiment, which may not be significant for practical purposes.

Commentary by Dr. Valentin Fuster
J. Fluids Eng. 1983;105(3):329-334. doi:10.1115/1.3240999.

The mixing length theory is extended to close the relevant momentum equations for two-phase turbulent flow at a first-order closure level. It is assumed that the mass fraction of the particles is on the order of unity, that the particle size is so small that the particles are fully suspended in the primary fluid, and that the relaxation time scale of the particles is sufficiently small compared with the time scale of the energy containing eddies so that the suspended particles are fully responsive to the fluctuating turbulent field. Bulk motion of the particles is treated as a secondary fluid flow with its own virtual viscosity. The proposed closure is applied to a fully developed gas-solid pipe flow in which the particles are assumed to be uniformly distributed across the pipe section. Predicted velocity profiles and the friction factors are in good agreement with available experimental data.

Commentary by Dr. Valentin Fuster
J. Fluids Eng. 1983;105(3):335-340. doi:10.1115/1.3241000.

Several attenuation configurations are assessed, involving the following concepts: generating streamwise vorticity, dephasing the azimuthal coherence of the jet, and reducing rigidity of the jet separation edge. By proper design, effective attenuation of the discrete frequency components of the oscillation can be achieved. For cases where there is not complete attenuation, the phase condition corresponding to maximum relative amplitude of the oscillation is maintained, the disturbance phase speed is essentially unaltered, and there is a proportional reduction in amplitude along the jet, including the initial fluctuation level at separation.

Commentary by Dr. Valentin Fuster
J. Fluids Eng. 1983;105(3):341-347. doi:10.1115/1.3241001.

An experimental study was made on the collapse of a gas bubble attached to a solid wall by a shock wave. The collapse process of the bubble and the induced impact wall pressure were measured simultaneously by means of a high speed camera and a pressure transducer, respectively. Consequently, it was found that the impact wall pressure was very sensitive to the factors such as the bubble size, the strength of shock wave and the distance from the origin of shock wave to the gas bubble, and in some cases it became larger than that generated by a shock wave directly impinging on the solid wall without a gas bubble.

Commentary by Dr. Valentin Fuster
J. Fluids Eng. 1983;105(3):350-355. doi:10.1115/1.3241004.

A “diffuser” is a venturi-like element with a well-formed contraction followed by a small-angle diffuser. When liquid flows through two diffusers connected in series various flow states are possible depending on whether or not cavitation occurs in the diffusers. It is shown that, in the absence of strong “Reynolds-number effects,” one of just two possible sequences of flow states can occur for a particular pair of diffusers. Denoting the diffusers by “upstream” and “downstream,” cavitation can occur as follows with increasing flow: 1. Neither; upstream only; both. 2. Neither; downstream only (and upstream never). Once this classification is known it is easy to predict the characteristics of the circuit which can then be used to define an “equivalent single diffuser” to represent the pair of diffusers. Experimental data verifying the theory are included.

Commentary by Dr. Valentin Fuster
J. Fluids Eng. 1983;105(3):356-362. doi:10.1115/1.3241005.

The method of matched asymptotic expansions is used to investigate the behavior of a collapsing bubble near a solid wall. Cases are studied in which the ratio ε between the initial spherical bubble radius and its distance from the wall is small. Expansions in powers of ε lead to a simple system of differential equations which is solved numerically. The bubble shape, the velocity potential and the pressure field are determined as functions of time. The deformation of the bubble is a singular perturbation of the pressure field around it. An increase in the value of ε augments the pressure on the solid wall by orders of magnitude. The influence of surface tension and the proximity of the wall, gas content and its law of compression, are investigated. The results are compared to previous investigations. One advantage of the method employed is the fact that it leads to a numerical solution which costs very little computer time. In addition, it can be extended very easily to more complex cases such as multibubble configurations or to walls coated with elastomeric coatings.

Commentary by Dr. Valentin Fuster



J. Fluids Eng. 1983;105(3):364-365. doi:10.1115/1.3241008.

A comparison of the Prandtl-Schlichting formula for skin friction of a fully rough plate with recently obtained experimental data shows an average error of 17.5 percent. It is suggested that the reason for this discrepancy is a failure to account for the wake component of the velocity profile. The integral momentum equation is used to derive a new skin friction theory which when compared to the same data gives an average error of 2.7 percent. A new skin friction formula is proposed which is valid over a wide parameter range.

Commentary by Dr. Valentin Fuster

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