Linearized k-ε Analysis of Free Turbulent Mixing in Streamwise Pressure Gradients With Experimental Verification

1979 ◽  
Vol 46 (3) ◽  
pp. 493-498 ◽  
Author(s):  
G. Hokenson
Keyword(s):  
Turbulent Mixing ◽  
Eddy Viscosity ◽  
Turbulence Modeling ◽  
Free Stream ◽  
Physical Space ◽  
Stream Velocity ◽  
Length Scale ◽  
Pressure Gradients ◽  
Preliminary Estimate ◽  
Closed Form Solutions

The equations of momentum, turbulent kinetic energy, and dissipation are subjected to a coordinate transformation and linearized to obtain approximate closed-form solutions of free mixing problems. The linearization involves not only an assumption regarding the relative transverse uniformity of free mixing flow fields, but also a turbulence modeling approach in which a preliminary estimate of the length scale is a necessary input. As a by-product of this linearization, the equations partially decouple from one another and may, therefore, be solved sequentially. In order to provide the length scale and free-stream velocity dependence upon the transformed streamwise coordinate, a temporary transformation from the physical to the mathematical plane is developed on the basis of a classical eddy viscosity formula. Due to the analytical nature of the process, the input velocity and length scale thus obtained may be adjusted to conform with the desired velocity distribution in physical space, and the appropriate length scale computed from the solution of the equations. The analysis is favorably compared to experimental data on the turbulent mixing of two-dimensional wakes in adverse pressure gradients.

Experiments on the flow past an inclined disk

1967 ◽  
Vol 29 (4) ◽  
pp. 691-703 ◽  
Author(s):  
J. R. Calvert
Keyword(s):  
Vortex Shedding ◽  
Free Stream ◽  
Stream Velocity ◽  
Length Scale ◽  
Angle Of Incidence ◽  
Axially Symmetric ◽  
Periodic Motions ◽  
Reference Velocity ◽  
Shedding Frequency ◽  
A Chain

The wake of a disk at an angle to a stream contains marked periodic motions which arise from the regular shedding of vortices from the trailing edge. The vortices are in the form of a chain of irregular rings, each one linked to the succeeding one, and they move downstream at about 0·6 of free-stream velocity. The prominence of the vortex shedding increases as the angle of incidence (measured from the normal) increases up to at least 50°. The shedding frequency increases with the angle of incidence, but by a suitable choice of reference velocity and length scale, may be described by a wake Strouhal number which has the constant value 0·21 for all angles of incidence above zero, up to at least 40°.Axially-symmetric bodies at zero incidence shed vortices in a similar manner, except that the orientation of the plane of vortex shedding is not fixed and varies from time to time.

Eddy Viscosity Distributions in Compressible Turbulent Boundary Layers with Injection

1971 ◽  
Vol 22 (2) ◽  
pp. 169-182 ◽  
Author(s):  
L. C. Squire
Keyword(s):  
Boundary Layer ◽  
Eddy Viscosity ◽  
Free Stream ◽  
Turbulent Boundary Layers ◽  
Stream Velocity ◽  
Mixing Length ◽  
Carbon Dioxide Injection ◽  
Displacement Thickness ◽  
Porous Surfaces ◽  
Injection Ratio

SummaryShear stress, eddy viscosity and mixing length distributions have been obtained from measured boundary-layer developments over porous surfaces with air and carbon dioxide injection at Mach numbers up to M=3·6. It is found that, if the eddy viscosity is non-dimensionalised by dividing by the product of the free-stream velocity and the kinematic displacement thickness then this non-dimensional ratio is almost independent of injection ratio, but decreases slightly with Mach number.

Turbulent Wakes in Pressure Gradients

1963 ◽  
Vol 30 (4) ◽  
pp. 518-524 ◽  
Author(s):  
P. G. Hill ◽  
U. W. Schaub ◽  
Y. Senoo
Keyword(s):  
Eddy Viscosity ◽  
Analytical Study ◽  
Positive Pressure ◽  
Stream Velocity ◽  
Moderate Pressure ◽  
Pressure Gradients ◽  
Momentum Integral ◽  
Theoretical Considerations ◽  
Acceptable Agreement ◽  
Made In

The application of moderate pressure gradients can exercise a large influence on the decay of wakes produced by obstructions in a flow field. In fact, a positive pressure gradient may arrest completely the decay process and cause the relative wake size to grow rapidly. This phenomenon could exist in any diffusing passage whose entrance flow has a wake-type distortion. Such a nonuniformity, due to a blade or a strut or other cause of stagnation-pressure variation in the stream, could lead to diffuser stall, as has been demonstrated experimentally. An analytical study of the problem has been made on the basis of measurements made in the M.I.T. Gas Turbine Laboratory. Theoretical considerations have led to a very convenient calculation formula which shows quite acceptable agreement with experimental data. The methods employ momentum integral equations and an “eddy viscosity” which is uniform across the wake and proportional to the product of the local free-stream velocity and the momentum thickness.

The Magnification of Roughness Drag by Pressure Gradients

1967 ◽  
Vol 71 (673) ◽  
pp. 44-46 ◽  
Author(s):  
J. F. Nash ◽  
P. Bradshaw
Keyword(s):  
Boundary Layer ◽  
Free Stream ◽  
Roughness Element ◽  
Dynamic Pressure ◽  
Subsequent Development ◽  
Adverse Pressure Gradient ◽  
Initial Boundary ◽  
Stream Velocity ◽  
Pressure Gradients ◽  
Roughness Elements

SummaryA simplified analysis indicates that the increase in profile drag of an aerofoil due to an isolated roughness element is, in general, different from the drag of the element measured on a flat plate with the same free-stream velocity. This “magnification” effect is caused chiefly by the effect of the pressure gradients on the boundary layer downstream of the roughness element.The degree of magnification is not closely approximated by the ratio of local to free-stream dynamic pressure and, in many typical cases, the contribution to the drag due to roughness elements may be seriously under-estimated in this way.Measurements of the effect of the initial boundary-layer thickness on the subsequent development of a turbulent boundary-layer in an adverse pressure gradient support the theoretical conclusions.

Bypass Transition Modelling: A New Method Which Accounts for Free-Stream Turbulence Intensity and Length Scale

2000 ◽  
Author(s):  
Paul E. Roach ◽  
David H. Brierley
Keyword(s):  
Boundary Layer ◽  
Turbulence Intensity ◽  
Free Stream ◽  
Leading Edge ◽  
Boundary Layer Transition ◽  
Test Surface ◽  
Length Scale ◽  
Pressure Gradients ◽  
Layer Transition ◽  
Free Stream Turbulence

The publication of the present authors’ boundary layer transition data in 1992 (now widely known as the ERCOFTAC test case T3) has led to a spate of new experimental and modelling efforts aimed at improving our understanding of this problem. This paper describes a new method of determining boundary layer transition with zero mean pressure gradient. The approach examines the development of a laminar boundary layer to the start of transition, accounting for the influences of free-stream turbulence and test surface geometry. It is presented as a “proof of concept”, requiring a significant amount of work before it can be considered as a practically applicable model for transition prediction. The method is based upon one first put forward by G.I. Taylor in the 1930’s, and accounts for the action of local, instantaneous pressure gradients on the developing laminar boundary layer. These pressure gradients are related to the intensity and length scale of turbulence in the free-stream using Taylor’s simple isotropic model. The findings demonstrate the need to account for the separate influences of free-stream turbulence intensity and length scale when considering the transition process. Although the length scale has less of an effect than the intensity, its influence is, nevertheless, significant and must not be overlooked. This fact goes a long way towards explaining the large scatter to be found in simple correlations which involve only the turbulence intensity. Intriguingly, it is demonstrated that it is the free-stream turbulence at the leading edge of the test surface which is important, not that found locally outside the boundary layer. The additional influence of leading edge geometry is also shown to play a major role in fixing the point at which transition begins. It is suggested that the leading edge geometry will distort the incident turbulent eddies, modifying the effective “free-stream” turbulence properties. Consequently, it is shown that the scale of the eddies relative to the leading edge thickness is a further important parameter, and helps bring together a large number of test cases.

Experiments on Leading-Edge Induced Separated Shear Layer Under Various Imposed Pressure Gradients

2014 ◽  
Author(s):  
K. Anand ◽  
S. Sarkar ◽  
N. Thilakan
Keyword(s):  
Reynolds Number ◽  
Shear Layer ◽  
Free Stream ◽  
Separation Bubble ◽  
Leading Edge ◽  
Free Stream Velocity ◽  
Stream Velocity ◽  
Pressure Gradients ◽  
Mean Flow ◽  
Separated Shear Layer

The behaviour of a separated shear layer past a semi-circular leading edge flat plate, its transition and reattachment downstream to separation are investigated for different imposed pressure gradients. The experiments are carried out in a blowing tunnel for a Reynolds number of 2.44×105 (based on chord and free-stream velocity). The mean flow characteristics and the instantaneous vector field are documented using a two-component LDA and a planar PIV, whereas, surface pressures are measured with Electronically scanned pressure (ESP). The onset of separation occurs near the blend point for all values of β (flap angle deflection), however, a considerable shift is noticed in the point of reattachment. The dimensions of the separation bubble is highly susceptible to β and plays an important role in the activity of the outer shear layer. Instantaneous results from PIV show a significant unsteadiness in the shear layer at about 30% of the bubble length, which is further amplified in the second half of the bubble leading to three-dimensional motions. The reverse flow velocity is higher for a favourable pressure gradient (β = +30°) and is found to be 21% of the free stream velocity. The Reynolds number calculated based on ll (laminar shear layer length), falls in the range of 0.9×104 to 1.4×104. The numerical values concerning the criterion for separation and reattachment agree well with the available literature.

scholarly journals The Turbulence That Matters

1997 ◽  
Author(s):  
R. E. Mayle ◽  
K. Dullenkopf ◽  
A. Schulz
Keyword(s):  
Heat Transfer ◽  
Reynolds Number ◽  
Gas Turbines ◽  
Free Stream ◽  
Transitional Flow ◽  
Stream Velocity ◽  
Length Scale ◽  
Effective Frequency ◽  
Large Wave ◽  
Free Stream Turbulence

A unified expression for the spectrum of turbulence is developed by asymptotically matching known expressions for small and large wave numbers, and a formula for the one-dimensional spectral function which depends on the turbulence Reynolds number Reλ is provided. In addition, formulas relating all the length scales of turbulence are provided. These relations also depend on Reynolds number. The effects of free-stream turbulence on laminar heat transfer and pre-transitional flow in gas turbines are re-examined in light of these new expressions using our recent thoughts on an ‘effective’ frequency of turbulence and an ‘effective’ turbulence level. The results of this are that the frequency most effective for laminar heat transfer is about 1.3U/Le, where U is the free-stream velocity and Le is the length scale of the eddies containing the most turbulent energy, and the most effective frequency for producing pre-transitional boundary layer fluctuations is about 0.3U/η where η is Kolmogorov’s length scale. In addition, the role of turbulence Reynolds number on stagnation heat transfer and transition is discussed, and new expressions to account for its effect are provided.

Free Turbulent Mixing in Axial Pressure Gradients

1973 ◽  
Vol 40 (2) ◽  
pp. 375-380 ◽  
Author(s):  
G. J. Hokenson ◽  
J. A. Schetz
Keyword(s):  
Flow Field ◽  
Turbulent Mixing ◽  
Eddy Viscosity ◽  
Static Pressure ◽  
Initial Conditions ◽  
Near Field ◽  
Pressure Gradients ◽  
Mean Flow ◽  
Axial Pressure ◽  
Flow Equations

The results of an experimental investigation of the free turbulent mixing of wakes and jets in axial pressure gradients are presented. The data include static pressure and velocity profiles and the turbulent intensity which is presented in terms of the parameter u′cL2¯(Δu)max2. It is hypothesized that the representation of the Reynolds stress by a generalized Clauser eddy viscosity model is scaled by this parameter. The experimentally observed dependence of this turbulence quantity on flow field dimensionality and the imposed pressure gradient places more stringent demands on the form of the eddy viscosity than has been shown before. However, the experimental data reveal some fortuitous behavior which aids in the specification of the spatial dependence of the turbulence parameter, leaving the scaling to be determined primarily by the initial conditions, i.e., the state of the turbulence in the near field. Substantial lateral static pressure gradients were observed in all two-dimensional cases studied. It is shown that the boundary-layer form of the viscous flow equations are inadequate in such cases, and a numerical solution of a system of equations that includes an approximate form of the lateral momentum equation provides predictions in good agreement with the data for the mean flow field.

Mixed Convection in Boundary Layer Flow on a Horizontal Plate

1977 ◽  
Vol 99 (1) ◽  
pp. 66-71 ◽  
Author(s):  
T. S. Chen ◽  
E. M. Sparrow ◽  
A. Mucoglu
Keyword(s):  
Heat Transfer ◽  
Free Stream ◽  
Stream Velocity ◽  
Local Similarity ◽  
Pressure Gradients ◽  
Local Surface ◽  
Flow And Heat Transfer ◽  
Buoyancy Forces ◽  
Layer Flow ◽  
Forced Convective Flow

The effects of buoyancy-induced streamwise pressure gradients on laminar forced convective flow and heat transfer over a horizontal flat plate are studied analytically by the local similarity and local nonsimilarity methods of solution. Numerical results for the local surface heat transfer, wall shear stress, and velocity and temperature distributions are presented for gases having a Prandtl number of 0.7. It is found that both the local Nusselt number and the friction factor increase with increasing buoyancy forces for aiding flow and decrease with increasing buoyancy forces for opposing flow. With regard to the heat transfer results, significant buoyancy effects were encountered for Grx/Rex5/2 > 0.05 and < −0.03, respectively, for aiding and opposing flows. The buoyancy-affected velocity profiles for the aiding-flow case exhibited an overshoot beyond the free stream velocity. Results from previously reported series solutions and from an integral momentum/energy solution were found to be accurate only when the buoyancy effects are small. The present study provides results for intermediate range buoyancy force effects, which have not been reported previously.

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