Turbulent flow in a conical diffuser: A review

This is a review of experimental studies of turbulent flow in a conical diffuser by eight Ph.D. students, eleven M.Sc. students, one M.Eng. student, and myself in the past 29 years. During this time, two conical diffusers were constructed: the first was of cast aluminum construction, and the second was of plastic fabrication. These two diffusers were basically the same in geometry except that the pipe section was constructed as an integral part of the plastic diffuser to avoid the lip at the junction of the inlet pipe and the diffuser. The conical diffuser had a total divergence angle of 8°, an area ratio of 4:1, and an inlet diameter of 0.1016 m (4 in.).

The flow at the inlet of the diffuser was usually fully developed pipe flow, but sometimes it was boundary layer grown on the pipe wall. Hot-wire and pulse-wire anemometry together with computer facilities were used to obtain the results of complex flow present in the conical diffuser. Mean velocity profiles were obtained throughout the diffuser, which in turn were used to obtain strain rates and their principal direction. Turbulence moments up to fourth order were measured. The results were used to assess momentum, turbulent kinetic energy, and shear stress equations. Other features such as instantaneous flow reversals in the wall region, relative strength of large eddies, extra strain rate, and the production of kinetic energy also were investigated to find the dynamical picture in the diffuser flow.     

Numerical prediction of turbulent flow in a conical diffuser usingk-ε model

Fully developed incompressible turbulent flow in a conical diffuser having a total divergence angle of 8° and an area ratio of 4∶1 has been simulated by ak-ε turbulence model with high Reynolds number and adverse pressure gradient. The research has been done for pipe entry Reynolds numbers of 1.16×10⁵ and 2.93×10⁵. The mean flow velocity and turbulence energy are predicted successfully and the advantage of Boundary Fit Coordinates approach is discussed. Furthermore, thek-ε turbulence model is applied to a flow in a conical diffuser having a total divergence angle of 30° with a perforated screen. A simplified mathematical model, where only the pressure drop is considered, has been used for describing the effect of the perforated screen. The optimum combination of the resistance coefficient and the location of the perforated screen is predicted for high diffuser efficiency or the uniform velocity distribution.     

Separation control in a conical diffuser with an annular inlet: center body wake separation

In many practical applications of conical diffusers, the flow is fed by an annular flow passage formed by a center body. Flow separation, which occurs if the center body ends abruptly, is undesirable because it degrades the diffuser performance. The present experiment utilizes magnetic resonance velocimetry to acquire three-component mean velocity measurements for a set of conical diffusers with an annular inlet. The results show strong coupling between the diffuser wall boundary layer development and the wake of the center body. Coanda blowing is used to mitigate the center body wake separation. The diffuser wall boundary layer is thick in the absence of the central separation bubble and separates when Coanda blowing is too strong.     

Slow flows of a viscous liquid in a conical diffuser

We show the applicability of Stokes' approximation at large distances from the vertex of a cone. We discuss the statement of the problem and formulate new asymptotic representations of the solution, which replace the paradoxical solution of Harrison for cone vertex angles α≥120°. A solution of the problem concerning the axially symmetric Stokes' flow of a viscous liquid in a conical diffuser was first obtained by Harrison [1] (see also [2, 3]). The velocity field of this flow has the form $$\upsilon _R = \frac{{3Q}}{{2\pi R^2 }}\frac{{\cos ^2 - \cos ^2 \alpha }}{{\left( {1 - \cos \alpha } \right)^2 \left( {1 + 2\cos \alpha } \right)}}, \upsilon _\theta = 0$$ where R and Θ are spherical coordinates, Θ=0 and Θ=α correspond, respectively, to the axis and to the wall of the diffuser, and Q is the volumetric outflow rate of the liquid. We note that the values of the velocity in this purely radial flow become infinite when the angle α approaches 120°.     

Near-wall velocity distributions within a straight conical diffuser

The measured mean velocity profiles at the various stations along a conical diffuser (8° total divergence angle) were found to consist of log regions, half-power law regions and linear regions. The describing coefficients for the inner half-power law region (which followed a rather narrow log region) differed from the standard values due to the axi-symmetric geometry and lack of moving equilibrium of the flow as it attempted to adjust to a varying adverse pressure gradient. However, these coefficients (like those for the linear region) correlated with the local wall shear stress and the kinematic pressure gradient.List of symbols A, B coefficients in logarithmic law velocity distribution (Eq. (1)) - C, D coefficients in half-power law velocity distribution (Eq. (5)) - D_i inside diameter of feed pipe (10.16 cm) - d _p outer diameter of Preston tube - E, F coefficients in linear law velocity distribution (Eq. (10)) - P _s local static pressure - R local radius of diffuser, (D _i /2) + x _w sin 4° - Re Reynolds number, D _i U _b /v - U local mean velocity in the x _w direction - U _b cross-sectional average mean velocity (x-direction) in feed pipe - U _c mean velocity at the diffuser centerline - u ^* local friction velocity - u ⁺ dimensionless local mean velocity, U/u ^* - axial distance along diffuser centerline (measured from inlet to diffuser) Fig. (2) - _w distance along diffuser wall (measured from inlet to difusser (Fig. 2) - y _w distance from wall in direction orthogonal to wall (Fig. 2) - y ⁺ dimensionless position, y _w u ^*/v - kinematic (axial static) pressure gradient, (1/g9) dP _s/dx - ^* displacement thickness (Eq. (4)) - dimensionless pressure gradient parameter, x v/(u*) ³ - Von Karman constant (0.41) - density - kinematic viscosity - shear stress     

Flow characteristics in the downstream region of a conical diffuser

Using the Navier-Stokes equations in conjunction with the k-? model of turbulence, the characteristics of flow in the region downstream of a conical diffuser with 5° angle of inclination are calculated. Two representative stations 1D₂ and 10D₂ after the diffuser exit are selected for comparison against experimental results. The calculations indicate an underestimation of mean velocity and turbulence kinetic energy at the first station, while satisfactory agreement is obtained for the mean velocity at the second station. The use of a modified k-? model sensitive to adverse pressure conditions improves the predictions considerably. The effect of inlet properties and Reynolds number on the flow characteristics at the above stations is studied using various inlet profiles and a range of Reynolds numbers based on the inlet diameter from 50 000 to 280 000.     

Oscillating flow in a 2-D diffuser

Separating oscillating flows in an internal, adverse pressure gradient geometry are studied experimentally. Simultaneous velocity and pressure measurements demonstrate that the minor losses associated with oscillating flow in an adverse pressure gradient geometry can be smaller or larger than those for steady flow. Separation is found to begin high in the diffuser and propagate downward. The flow is able to remain attached further into the diffuser with larger Reynolds numbers, small displacement amplitudes, and smaller diffuser angles. The extent of separation grows with L ₀/h. The minor losses grow with increasing displacement amplitude in the measured range 10 < L ₀/h < 40. Losses decrease with increasing Re _δ in the measured range of 380 < Re _δ < 740. It is found that the losses increase with increasing diffuser angle over the measured range of 12° < θ < 30°. The nondimensional acoustic power dissipation increases with Reynolds number in the measured range and decreases with displacement amplitude.     

Flow of a viscous liquid in a conical diffuser in the presence of heat transfer

In this work a solution has been derived for the motion of an incompressible liquid with a temperature dependent viscosity in a conical diffuser. The inertia and diffusion terms are neglected in the equations of motion and heat conduction. It is assumed that the temperature of the diffuser wall is inversely proportional to the spherical radius. It is shown that the stream function and temperature are uniformly convergent series, the terms of which satisfy an infinite system of normal differential equations. An investigation has been made of the behavior of the flow pattern and the effects of a temperature gradient on the radial velocity component. The method has been used to resolve the flow of a liquid between two coaxial cones.     

Turbulent separated reattached flow in a two-dimensional curved-wall diffuser

A turbulent separation-reattachment flow in a two-dimensional asymmetrical curved-wall diffuser is studied by a two-dimensional laser doppler velocimeter. The turbulent boundary layer separates on the lower curved wall under strong pressure gradient and then reattaches on a parallel channel. At the inlet of the diffuser, Reynolds number based on the diffuser height is 1.2×10⁵ and the velocity is 25.2m/s. The results of experiments are presented and analyzed in new defined streamline-aligned coordinates. The experiment shows that after Transitory Detachment Reynolds shear stress is negative in the near-wall backflow region. Their characteristics are approximately the same as in simple turbulent shear layers near the maximum Reynolds shear stress. A scale is formed using the maximum Reynolds shear stresses. It is found that a Reynolds shear stress similarity exists from separation to reattachment and the Schofield-Perry velocity law exists in the forward shear flow. Both profiles are used in the experimental work that leads to the design of a new eddy-viscosity model. The length scale is taken from that developed by Schofield and Perry. The composite velocity scale is formed by the maximum Reynolds shear stress and the Schofield-Perry velocity scale as well as the edge velocity of the boundary layer. The results of these experiments are presented in this paper.     

Formation of multiple shocklets in a transonic diffuser flow

Multiple shocklets are frequently generated in transonic diffuser flows. The present paper investigates the formation of these shocklets with a high-speed CCD camera combined with the schlieren method. It is observed that compression waves steepen while propagating upstream, and eventually become new shock waves. The ordinary shock wave is found to move upstream beyond the nozzle throat or to disappear while moving downstream depending on the pressure ratio across the nozzle. This phenomenon is also analyzed with the one-dimensional Euler equations by assuming a pressure disturbance given by the sine function at the channel exit. The calculated results are found to reproduce quite well the experimental behavior of the shocklets. The effect of the frequency of disturbance is also studied numerically, and it is shown that the multiple shocklet pattern appears when the amplitude of disturbance is not large and the diverging part of the channel downstream of the ordinary shock wave is long. Received 26 June 1998 / Accepted 15 March 1999     

跨超声速风洞扩散段流动分离控制设计参数研究

为抑制跨超声速风洞扩散段的分离,提出了一种较为完备的设计方法。由于影响扩散段性能的参数较多,完全通过试验方法进行设计的成本过高,该方法通过数值模拟,结合适当的边界条件,详细描述了扩散段角度、分流锥角度与长度、孔板开孔率对扩散段性能的影响;从数值模拟的结果可以看出,孔板开孔率和扩开角对扩散段性能有显著影响,通过比较得出较为合理的参数匹配,提高了扩散段的防分离性能,并改善了出口气流质量。数值结果与试验结果结论一致,表明本文所用的方法用于扩散段气动设计是可行的,为数值模拟方法应用于风洞部段气动设计创造了一定的条件。     

Effects of sound on flow separation from blunt flat plates

The effect of sound on the flow around plates with semicircular or square leading edges and square trailing edges located in a low turbulence open jet has been studied. In all circumstances the length of the leading edge separation bubbles associated with square leading edge plates was found to oscillate. When sound was applied to the flow around these plates, the leading edge shear layers reattached closer to the leading edge and the oscillations in bubble length occurred at the applied sound frequency, generating patches of concentrated vorticity in the boundary layers. These vorticity patches moved downstream near the plate surface and then beyond the trailing edge to form vortex cores in a street with a Strouhal number equal to the applied sound value. Sometimes these vortex streets are unstable and break down into streets with Strouhal numbers approaching those observed without sound. These effects of sound were not observed in the flow around plates with semicircular leading edges. Without sound, square leading edge plates of intermediate length did not shed regular vortex streets.     

Nonisothermal flow of a viscous incompressible fluid in a two-dimensional diffuser

The velocity and temperature distributions in a viscous incompressible fluid flow in a two-dimensional diffuser are analyzed. Fully developed flow is considered, i.e., the influence of the entrant section is disregarded. It is assumed that the diffuser walls are maintained at a temperature depending on the polar radius. The dynamic viscosity is considered to be an exponential function of the temperature. The problem is reduced to the solution of a system of ordinary differential equations, which is solved by the method of successive approximations. The convergence of the iterative scheme is proved.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 40–48, July–August, 1973.The author is indebted to L.A. Galin and N. N. Gvozdkov for assistance with the study.     

Characteristics of aerodynamic sound sources generated by coiled wires in a uniform air-flow

This study deals experimentally with aerodynamic sounds generated by coiled wires in a uniform air-flow. The coiled wire is a model of the hair dryer's heater. In the experiment, the effects of the coil diameter D, wire diameter d and coil spacing s of the coiled wire on the aerodynamic sound have been clarified. The results of frequency analyses of the aerodynamic sounds show that an Aeolian sound is generated by the coiled wire, when s/d is larger than 1. Also the peak frequencies of Aeolian sounds generated by the coiled wires are higher than the ones generated by a straight cylinder having the same diameter d. To clarify the characteristics of the aerodynamic sound sources, the directivity of the aerodynamic sound generated by the coiled wire has been examined, and the coherent function between the velocity fluctuation around the coiled wire and the aerodynamic sound has been calculated. Moreover, the band overall value of coherent output power between the sound and the velocity fluctuations has been calculated. This method has clarified the sound source region of the Aeolian sound generated by the coiled wire. These results show that the Aeolian sound is generated by the arc part of the coiled wire, which is located in the upstream side of the air-flow.     

Measurements in the vaneless diffuser of a radial flow compressor

Results from an experimental study of flow behaviour at the inlet of a vaneless diffuser of a centrifugal compressor are presented. Measurements from a crossed hot-wire probe are given for operating points having inlet flow coefficients ranging from 0.006 to 0.019 at different Reynolds numbers. Instantaneous, time-averaged, and phase-averaged absolute velocity and flow angle at the diffuser inlet are deduced from the hot-wire signals after correction for mean density variations. These results show how flow behaviour varies in stable, rotating stall and surge regimes of compressor operation     

Low frequency scattering by a finite cone

The scattering of a low frequency electromagnetic wave by a metallic cone, whose base is part of a spherical surface centered on the apex of the cone, is analyzed using a mode matching technique. The dipole contributions to the scattering are obtained in complete generality, and numerical results are presented for a wide range of cone angles. Comparisons of the computed data with the predictions of an empirical formula for the scattering reveal both the strengths and weaknesses of the latter.This work was supported in part by the National Science Foundation under Grant GP 9642.     

Centreline velocity decay characterisation in low-velocity jets downstream from an extended conical diffuser

The centreline velocity decay of round airflow jets issuing from extended conical diffusers with length-to-diameter ratio 1.2≤L _t /d≤20 is studied for moderate bulk Reynolds numbers 1131≤Re _b ≤9054. The centreline velocity decay varies as a function of the initial conditions. The functional correlation between the centreline velocity decay coefficient and the initial centreline turbulence level observed on convergent nozzles (Malmström et al. in J. Fluid Mech. 246:363–377, 1997) breaks down as the initial centreline turbulence level exceeds 20 %. In addition, the centreline velocity decay coefficient expressed as function of the bulk velocity U _b decreases for U _b <3 m/s instead of initial mean velocity U ₀<6 m/s as reported for convergent nozzles (Malmström et al. in J. Fluid Mech. 246:363–377, 1997). The asymptotic values of the decay coefficient for U _b >3 m/s decrease linearly when expressed as function of the initial centreline turbulence intensity u ₀/U ₀. Studied flow and geometrical conditions are relevant to flow through the human upper airways.