The unsteady laminar boundary layer on a semi-infinite flat plate due to small fluctuations in the magnitude of the free-stream velocity

1972 ◽  
Vol 51 (1) ◽  
pp. 137-157 ◽  
Author(s):  
R. C. Ackerberg ◽  
J. H. Phillips

Asymptotic and numerical solutions of the unsteady boundary-layer equations are obtained for a main stream velocity given by equation (1.1). Far downstream the flow develops into a double boundary layer. The inside layer is a Stokes shear-wave motion, which oscillates with zero mean flow, while the outer layer is a modified Blasius motion, which convects the mean flow downstream. The numerical results indicate that most flow quantities approach their asymptotic values far downstream through damped oscillations. This behaviour is attributed to exponentially small oscillatory eigenfunctions, which account for different initial conditions upstream.

The boundary-layer equations for a compressible fluid are transformed into those for an incompressible fluid, assuming that the boundary is thermally insulating, that the viscosity is proportional to the absolute temperature, and that the Prandtl number is unity. Various results in the theory of incompressible boundary layers are then taken over into the compressible theory. In particular, the existence of ‘similar’ solutions is proved, and Howarth’s method for retarded flows is applied to determine the point of separation for a uniformly retarded main stream velocity. A comparison with an exact solution is used to show that this method gives a closer approximation than does Pohlhausen’s.


2015 ◽  
Vol 783 ◽  
pp. 379-411 ◽  
Author(s):  
I. Marusic ◽  
K. A. Chauhan ◽  
V. Kulandaivelu ◽  
N. Hutchins

In this paper we study the spatial evolution of zero-pressure-gradient (ZPG) turbulent boundary layers from their origin to a canonical high-Reynolds-number state. A prime motivation is to better understand under what conditions reliable scaling behaviour comparisons can be made between different experimental studies at matched local Reynolds numbers. This is achieved here through detailed streamwise velocity measurements using hot wires in the large University of Melbourne wind tunnel. By keeping the unit Reynolds number constant, the flow conditioning, contraction and trip can be considered unaltered for a given boundary layer’s development and hence its evolution can be studied in isolation from the influence of inflow conditions by moving to different streamwise locations. Careful attention was given to the experimental design in order to make comparisons between flows with three different trips while keeping all other parameters nominally constant, including keeping the measurement sensor size nominally fixed in viscous wall units. The three trips consist of a standard trip and two deliberately ‘over-tripped’ cases, where the initial boundary layers are over-stimulated with additional large-scale energy. Comparisons of the mean flow, normal Reynolds stress, spectra and higher-order turbulence statistics reveal that the effects of the trip are seen to be significant, with the remnants of the ‘over-tripped’ conditions persisting at least until streamwise stations corresponding to $Re_{x}=1.7\times 10^{7}$ and $x=O(2000)$ trip heights are reached (which is specific to the trips used here), at which position the non-canonical boundary layers exhibit a weak memory of their initial conditions at the largest scales $O(10{\it\delta})$, where ${\it\delta}$ is the boundary layer thickness. At closer streamwise stations, no one-to-one correspondence is observed between the local Reynolds numbers ($Re_{{\it\tau}}$, $Re_{{\it\theta}}$ or $Re_{x}$ etc.), and these differences are likely to be the cause of disparities between previous studies where a given Reynolds number is matched but without account of the trip conditions and the actual evolution of the boundary layer. In previous literature such variations have commonly been referred to as low-Reynolds-number effects, while here we show that it is more likely that these differences are due to an evolution effect resulting from the initial conditions set up by the trip and/or the initial inflow conditions. Generally, the mean velocity profiles were found to approach a constant wake parameter ${\it\Pi}$ as the three boundary layers developed along the test section, and agreement of the mean flow parameters was found to coincide with the location where other statistics also converged, including higher-order moments up to tenth order. This result therefore implies that it may be sufficient to document the mean flow parameters alone in order to ascertain whether the ZPG flow, as described by the streamwise velocity statistics, has reached a canonical state, and a computational approach is outlined to do this. The computational scheme is shown to agree well with available experimental data.


1996 ◽  
Vol 309 ◽  
pp. 157-182 ◽  
Author(s):  
T. B. Nickels ◽  
A. E. Perry

Experiments carried out recently on a family of axisymmetric coflowing turbulent jets with different nozzle to free-stream velocity ratios are described. Special care was taken to ensure top-hat velocity profiles at the nozzle exit so as to reduce the number of parameters associated with the initial conditions. This results in a collapse of the data without the need to introduce different effective origins for the streamwise coordinate. The mean flow behaviour is compared to self-preserving asymptotic forms and stresses are also examined to investigate the possibility of self-preservation. Comparisons are made with measurements of other workers in coflowing jets and axisymmetric wakes. Further information on the structure of the coflowing jet is found by examining the spectral Reynolds shear stress correlation coefficient, the premultiplied spectra and the high-wavenumber forms of the spectra. An analysis was carried out to see if the mean flow, Reynolds stress distributions and spectra are consistent with an inviscid ‘double-roller’ vortex structure for the representative large-scale energy-containing motions. Results show support for such a model.


Author(s):  
J. C. Burns

ABSTRACTClassical shallow-water theory for the propagation of long waves in running water is modified by the inclusion of the effects of the vorticity present in the main stream as the result of the action of viscosity. When this vorticity is assumed constant, a non-linear theory can be used, but for more general velocity distributions in the main stream it is necessary to linearize the problem.In the linearized theory, a general equation is obtained connecting the wave velocity with the velocity in the undisturbed stream and this is solved in several special cases. It is shown generally that the wave velocity relative to the mean flow is always greater than the value given by the classical theory. The wave velocity relative to the bottom of the stream has two values, one less than the minimum stream velocity and the other greater than the maximum stream velocity.


Author(s):  
Heath Chalmers ◽  
Xingjun Fang ◽  
Mark F. Tachie

Abstract Separated and reattached turbulent flows induced by two-dimensional forward-backward-facing steps with different streamwise lengths submerged in a thick turbulent boundary layer are investigated using a time-resolved particle image velocimetry. The examined aspect ratios of the step range from 1 to 8, and the Reynolds number based on the free-stream velocity and step height is 13 200. The thickness of the oncoming turbulent boundary layer is 6.5 times the step height. The effects of varying aspect ratio of the steps on the mean flow, Reynolds stresses, triple correlations and unsteadiness of turbulent separation bubbles are studied. It was found that the mean flow reattaches over the step for forward-backward facing steps with aspect ratios of 2 and higher. The temporal variation of the first proper orthogonal decomposition (POD) mode and reverse flow area, which is used to examine the flapping motion of separation bubble, shows remarkable synchronization.


This paper describes a combined theoretical and experimental investigation into the heat transfer from a disk rotating close to a stator with a radial outflow of coolant. Experimental results are obtained from a 762 mm diameter disk, rotating up to 4000 rev/min at axial clearances from 2 to 230 mm from a stator of the same diameter, with coolant flow rates up to 0.7 kg/s. Mean Nusselt numbers are presented for the free disk, the disk rotating close to an unshrouded stator with no coolant outflow, the disk rotating close to a shrouded and unshrouded stator with coolant outflow, and for the unshrouded stator itself. Numerical solutions of the turbulent boundary layer equations are in satisfactory agreement with the experimentally determined mean Nusselt numbers for the air-cooled disk over a wide range of conditions. At large ratios of mass flow rate/rotational speed the mean Nusselt numbers for the air-cooled disk are independent of rotation, and both the numerical solutions and experimental results become asymptotic to an approximate solution of the boundary layer equations.


2021 ◽  
Author(s):  
Samuel Addai ◽  
Afua A. Mante ◽  
Sedem Kumahor ◽  
Xingjun Fang ◽  
Mark F. Tachie

Abstract In the present study, the effects of wall proximity on the wake dynamics behind a square cylinder subjected to a thick upstream turbulent boundary layer were experimentally investigated using particle image velocimetry. The Reynolds number based on the free-stream velocity and the cylinder height (h) was 12750 while the ratio of the turbulent boundary layer thickness to the cylinder height was 3.6. The gap distance (G) between the bottom face of the cylinder and the wall was varied, resulting in gap ratios (G/h) of 0, 0.3, 0.5, 1.0, 2.0 and 8.0. The flow topological differences among the various gap ratios were analyzed in terms of the mean flow and Reynolds stresses. The results show that as the cylinder approaches the wall, the mean flow becomes increasingly asymmetric about the horizontal centerline of the cylinder and the size of the mean separation bubbles in the cylinder wake increases. Also, the magnitudes of the Reynolds stresses decrease with decreasing gap ratio. For G/h > 0, the distributions of the streamwise Reynolds normal stress and Reynolds shear stress are concentrated along the upper and lower separated shear layers, resulting in characteristic double peaks. The distributions of the vertical Reynolds normal stress, however, are concentrated in the wake about the horizontal centerline of the cylinder and reveal only single peaks.


Author(s):  
D. Furey ◽  
P. Atsavapranee ◽  
K. Cipolla

Stereo Particle Image velocimetry data was collected over high aspect ratio flexible cylinders (L/a = 1.5 to 3 × 105) to evaluate the axial development of the turbulent boundary layer where the boundary layer thickness becomes significantly larger than the cylinder diameter (δ/a>>1). The flexible cylinders are approximately neutrally buoyant and have an initial length of 152 m and radii of 0.45 mm and 1.25 mm. The cylinders were towed at speeds ranging from 3.8 to 15.4 m/sec in the David Taylor Model Basin. The analysis of the SPIV data required a several step procedure to evaluate the cylinder boundary flow. First, the characterization of the flow field from the towing strut is required. This evaluation provides the residual mean velocities and turbulence levels caused by the towing hardware at each speed and axial location. These values, called tare values, are necessary for comparing to the cylinder flow results. Second, the cylinder flow fields are averaged together and the averaged tare fields are subtracted out to remove strut-induced ambient flow effects. Prior to averaging, the cylinder flow fields are shifted to collocate the cylinder within the field. Since the boundary layer develops slowly, all planes of data occurring within each 10 meter increment of the cylinder length are averaged together to produce the mean boundary layer flow. Corresponding fields from multiple runs executed using the same experimental parameters are also averaged. This flow is analyzed to evaluate the level of axisymmetry in the data and determine if small changes in cylinder angle affect the mean flow development. With axisymmetry verified, the boundary flow is further averaged azimuthally around the cylinder to produce mean boundary layer profiles. Finally, the fluctuating velocity levels are evaluated for the flow with the cylinder and compared to the fluctuating velocity levels in the tare data. This paper will first discuss the data analysis techniques for the tare data and the averaging methods implemented. Second, the data analysis considerations will be presented for the cylinder data and the averaging and cylinder tracking techniques. These results are used to extract relevant boundary layer parameters including δ, δ* and θ. Combining these results with wall shear and momentum thickness values extracted from averaged cylinder drag data, the boundary layer can be well characterized.


1988 ◽  
Vol 186 ◽  
pp. 583-597 ◽  
Author(s):  
P. M. Eagles

We find certain exact solutions of Jeffery-Hamel type for the boundary-layer equations for film flow over certain beds. If β is the angle of the bed with the horizontal and S is the arclength these beds have equation sin β = (const.)S−3, and allow a description of flows on concave and convex beds. The velocity profiles are markedly different from the semi-Poiseuille flow on a plane bed.We also find a class of beds in which the Jeffery-Hamel flows appear as a first approximation throughout the flow field, which is infinite in streamwise extent. Since the parameter γ specifying the Jeffery-Hamel flow varies in the streamwise direction this allows a description of flows over curved beds which are slowly varying, as described in the theory, in such a way that the local approximation is that Jeffery-Hamel flow with the local value of γ. This allows the description of flows with separation and reattachment of the main stream in some cases.


This paper contains a study of the similarity solutions of the boundary layer equations for the case of strong blowing through a porous surface. The main part of the boundary layer is thick and almost inviscid in these conditions, but there is a thin viscous region where the boundary layer merges into the main stream. The asymptotic solutions appropriate to these two regions are matched to one another when the blowing velocity is large. The skin friction is found from the inner solution, which is independent of the outer solution, but the displacement thickness involves both solutions and is of more complicated form.


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