Incompressible Mixing Research Articles

Characterization of the organization in shear layers via the Proper Orthogonal Decomposition

Experiments are performed in an incompressible plane turbulent mixing layer, using various hot wire rake configurations. From these experiments, the Proper Orthogonal Decomposition is applied for kernels where the space-time correlation tensor is evaluated over different spatial meshes and velocity components configurations. The resulting decompositions are then discussed in terms of “characterization of the organization of the flow” for various scalar or vectorial approaches of the POD. An incrtial range law is evidenced. The instantaneous contribution of the first modes of the POD to the organization of the flow is analyzed. A dynamical behavior for the organization of the flow is observed from the correlation between the first two modes contribution.

Early-period dynamics of an incompressible mixing layer

The evolution of three-dimensional disturbances in an incompressible mixing layer in an inviscid fluid is investigated as an initial-value problem. A Green's function approach is used to obtain a general space–time solution to the problem using a piecewise linear model for the basic flow, thereby making it possible to determine complete and closed-form analytical expressions for the variables with arbitrary input. Structure, kinetic energy, vorticity, and the evolution of material particles can be ascertained in detail. Moreover, these solutions represent the full three-dimensional disturbances that can grow exponentially or algebraically in time. For large time, the behaviour of these disturbances is dominated by the exponentially increasing discrete modes. For the early time, the behaviour is controlled by the algebraic variation due to the continuous spectrum. Contrary to Squire's theorem for normal mode analysis, the early-time behaviour indicates growth at comparable rates for all values of the wavenumbers and the initial growth of these disturbances is shown to rapidly increase. In particular, the disturbance kinetic energy can rise to a level approximately ten times its initial value before the exponentially growing normal mode prevails. As a result, the transient behaviour can trigger the roll-up of the mixing layer and its development into the well-known pattern that has been observed experimentally.

A study of the fine-scale motions of incompressible time-developing mixing layers

The geometry of dissipating motions in direct numerical simulations (DNS) of the incompressible mixing layer is examined. All nine partial derivatives of the velocity field are determined at every grid point in the flow, and various invariants and related quantities are computed from the velocity gradient tensor. Motions characterized by high rates of kinetic energy dissipation and high enstrophy density are of particular interest. Scatter plots of the invariants are mapped out and interesting and unexpected patterns are seen. Depending on initial conditions, each type of shear layer produces its own characteristic scatter plot. In order to provide more detailed information on the distribution of invariants at intermediate and large scales, scatter plots are replaced with more useful number density contour plots. These essentially represent the unnormalized joint probability density function of the two invariants being cross-plotted. Plane mixing layers at the same Reynolds number, but with laminar and turbulent initial conditions, are studied, and comparisons of the rate-of-strain topology of the dissipating motions are made. The results show conclusively that, regardless of initial conditions, the bulk of the total kinetic energy dissipation is contributed by intermediate scale motions, whose local rate-of-strain topology is characterized as unstable-node-saddle–saddle (two positive rate-of-strain eigenvalues, one negative). In addition, it is found that, for these motions, the rate-of-strain invariants tend to approximately follow a straight line relationship, characteristic of a two-dimensional flow with out of plane straining. In contrast, fine-scale motions, which have the highest dissipation, but which only contribute a small fraction of the total dissipation tend toward a fixed ratio of the principal rates of strain.

LDV Investigations of Supersonic Turbulent Mixing Layer. (Single-Stream Seeding Experiments).

Turbulence characteristics of the supersonic turbulent mixing layer were investigated using a two-component LDV. The high-speed stream was of Mach number M1= 2.24 and low-speed one was of M2=1.61 (convective Mach number Mc=0.18, and free-stream velocity ratio r=0.86). To examine the behavior of fluid entrained into the mixing layer, single-stream seeding experiments were conducted ; either a high-speed stream or low-speed one was seeded. In the fully developed regions of the mixing layer, it was found that the mean streamwise velocity of fluid within the mixing layer, which originated from the high-speed stream, was not equal to that of the fluid from the low-speed stream. The transverse locations in which the turbulence intensity and Reynolds stress showed maximum values for the high-speed-stream seeding case was not coincident with that of the low-speed seeding case. They were also compared with that of the incompressible mixing layer case. The growth rate of the mixing layer was almost same as that of the incompressible case. The mixing length based on Prandtl's mixing length hypothesis was also discussed.

Mean and turbulent velocity measurements of supersonic mixing layers

The behavior of supersonic mixing layers under three conditions has been examined by schlieren photography and laser Doppler velocimetry. In the schlieren photographs, some largescale, repetitive patterns were observed within the mixing layer; however, these structures do not appear to dominate the mixing layer character under the present flow conditions. It was found that higher levels of secondary freestream turbulence did not increase the peak turbulence intensity observed within the mixing layer, but slightly increased the growth rate. Higher levels of freestream turbulence also reduced the axial distance required for development of the mean velocity. At higher convective Mach numbers, the mixing layer growth rate was found to be smaller than that of an incompressible mixing layer at the same velocity and freestream density ratio. The increase in convective Mach number also caused a decrease in the turbulence intensity (σ u /ΔU).

Coherent Structures of a Supersonic Mixing Layer by Direct Numerical Simulations.

In the present paper, coherent structures of the supersonic mixing layer are investigated by direct numerical simulations of a spatialy developing free shear layer, We examined numerical results of the velocity components, u, v, w simulated by Fouillet and Lesieur : Reynolds number Re=δω0ΔU/ν=1500, Mach number =1(M1=1 and Mc=0.3), (U1+U2)/2=0.7, U1-U2=0.6, and the hyperbolic-tangent velocity profile is perturbed by weak three-dimensional white noise. We find that the coherent structures in the supersonic mixing layer at Mc=0.3 are three dimensional and have rib structures with strong vorticities of ωy and ωx similar to those in the incompressible mixing layer. The helical pairing structures found in the rib structure generate a coherent 'upwash' and 'downwash' fluid motion in the supersonic mixing layer. The regions corresponding to high rates of dissipation are found in the rib structures.

Inviscid instability of a skewed compressible mixing layer

In this paper we study the inviscid instability of a skewed compressible mixing layer between streams of different velocity magnitude and direction. The mean flow is governed by the three-dimensional laminar boundary-layer equations and can be reduced to a sum of a uniform flow and a two-dimensional shear flow. In the stability analysis, the amplification direction is assumed to be normal to the homogeneous direction of the mean flow. The results show that skewing enhances the instability by a factor of three for the incompressible mixing layer with velocity ratio 0.5 and uniform temperature. Under compressible conditions, skewing still increases the maximum amplification rate for a medium convective Mach number, but the enhancement is smaller. A scaling of the skewing effect is introduced which quantitatively explains the linear stability behaviour. Similarly, a suitably defined convective Mach number explains the compressibility effect.

Frequency spectra of reactant fluctuations in turbulent flows

Bilger, Saetran & Krishnamoorthy (1991) give measured values of the variance, cross-correlation coefficient, autospectra, coherence and phase shift of the reactant concentration fluctuations for an irreversible second-order reaction in an incompressible turbulent scalar mixing layer. The present paper approaches the interpretation of the measured data by evaluating the above quantities in the frozen (slow) and equilibrium (fast) chemistry limits. We assume that the limiting values bracket the corresponding intermediate rate data.The analysis leads to values that correspond with the measured variances and correlation coefficients. The paper offers simple procedures for experimenters to evaluate the fast chemistry limit of the spectral characteristics from the measured mixture fraction fluctuations. The investigation of the limiting spectra suggests that, in the frequency region considered in the Bilger et al. measurements, the shape of the autospectrum is quite insensitive to the chemistry rate. The cross-spectrum is much more sensitive to the chemistry than the autospectrum. The analysis predicts correctly that the coherence decreases with increasing frequency while the phase stays equal to π until the decrease of the coherence leads to indeterminate phase results.

Weakly nonlinear models for turbulent mixing in a plane mixing layer

New closure models for turbulent free shear flows are presented in this paper. They are based on a weakly nonlinear theory with a description of the dominant large-scale structures as instability waves. Two models are presented that describe the evolution of the free shear flows in terms of the time-averaged mean flow and the dominant large-scale turbulent structure. The local characteristics of the large-scale motions are described using linear theory. Their amplitude is determined from an energy integral analysis. The models have been applied to the study of an incompressible mixing layer. For both models, predictions of the mean flow development are made. In the second model, predictions of the time-dependent motion of the large-scale structures in the mixing layer are made. The predictions show good agreement with experimental observations.

Investigation of supersonic mixing layers

A constructed, dual-stream, supersonic wind tunnel has been designed to provide comparatively long run times for investigations of reactive and nonreactive, two-dimensional supersonic mixing layers. Control of the stagnation temperatures enables the velocity ratio, density ratio, and convective Mach number of the two streams to be varied. Laser Doppler velocimetry has been employed to measure the incoming boundary layers, the subsequent compressible mixinglayer growth rate, and the mean and turbulent velocity field development for two density ratios of 0.76 and 1.56, corresponding to velocity ratios of 0.79 and 0.58 and convective Mach numbers of 0.20 and 0.45, respectively. For a given density and velocity ratio, the convective Mach number has been found to correlate the ratio of compressible to incompressible mixing layer growth rate, as suggested by other workers.

Mean and turbulent velocity measurements of supersonic mixing layers

The behavior of supersonic mixing layers under three conditions has been examined by schlieren photography and laser Doppler velocimetry. In the schlieren photographs, some large-scale, repetitive patterns were observed within the mixing layer; however, these structures do not appear to dominate the mixing layer character under the present flow conditions. It was found that higher levels of secondary freestream turbulence did not increase the peak turbulence intensity observed within the mixing layer, but slightly increased the growth rate. Higher levels of freestream turbulence also reduced the axial distance required for development of the mean velocity. At higher convective Mach numbers, the mixing layer growth rate was found to be smaller than that of an incompressible mixing layer at the same velocity and freestream density ratio. The increase in convective Mach number also caused a decrease in the turbulence intensity (σu/ΔU).

Axially symmetrical jet mixing of an incompressible dusty fluid

Incompressible laminar jet mixing of a dustry fluid issuing from a circular jet has been considered by using Saffman's model for dusty fluid. Assuming the velocity and temperature in the jet to differ only slightly from that of the surrounding stream, a perturbation method has been employed to linearize the basic equations. The linearized equations have been solved by using Hankel transform, technique. Numerical computations have been made to discuss the veoocity and temperature profiles. It is observed that the particles move faster than the fluid in the down stream direction and the particles temperature is lower than the fluid temperature.

A statistical model of turbulence in two-dimensional mixing layers

A statistical model based on the proposition that the turbulence of a fully developed two-dimensional incompressible mixing layer is in a state of quasi-equilibrium is developed. In this model the large structures observed by Brown & Roshko (1974) which will be assumed to persist into the fully developed turbulent region are represented by a superposition of the normal wave modes of the flow with arbitrary random amplitudes. The turbulence at a point in the flow is assumed to be dominated by the fluctuations associated with these large structures. These structures grow and amalgamate as they are convected in the flow direction. Because of the lack of intrinsic length and time scales the turbulence in question can, therefore, be regarded as created or initiated at an upstream point, the virtual origin of the mixing layer, by turbulence with a white noise spectrum and are subsequently convected downstream. The model is used to predict the second-order turbulence statistics of the flow including single point turbulent Reynolds stress distribution, intensity of turbulent velocity components, root-mean-square turbulent pressure fluctuations, power spectra and two-point space-time correlation functions. Numerical results based on the proposed model compare favourably with available experimental measurements. Predictions of physical quantities not yet measured by experiments, e.g. the root-mean-square pressure distribution across the mixing layer, are also made. This permits the present model to be further tested experimentally.

On the two-dimensional mixing region

An experimental investigation of the two-dimensional incompressible mixing layer was carried out. The measurements provide new information on the development of the mean and turbulent fields towards a self-preserving state and on the higher-order statistical characteristics of the turbulent field. The relevance of initial conditions to the development of the flow is discussed in the light of both present and previous data. Measurements of spectra, probability densities and moments to eighth order of all three velocity-component fluctuations at various transverse positions across the flow were carried out using an on-line digital data acquisition system. The probability density distributions of the derivative and the squared derivative of the longitudinal and lateral velocity fluctuations were also determined. Direct measurements of moments to eighth order of the velocity derivatives were attempted and are discussed in the light of the simultaneously measured histograms. The problems in obtaining higher-order statistical data are considered in some detail. Estimates of the integral time scale of many of the higher-order statistics are presented. The high wave-number structure was found to be locally anisotropic according to both spectral and turbulent velocity-gradient moment requirements. Higher-order spectra to fourth order of the longitudinal velocity fluctuations were measured and are discussed. Finally the lognormality of the squared longitudinal and lateral velocity-derivative fluctuations was investigated and the universal lognormal constant μ was evaluated.

The maintenance of turbulent shear stress in a mixing layer

Wavenumber frequency spectra have been measured in a two-dimensional incompressible mixing layer, using linearized hot-wire anemometers. Spectra of two dimensions (frequency and wavenumber) have been measured for lateral and longitudinal turbulent velocities, and used to construct three-dimensional spectra. The validity of the separation assumption used to construct these spectra was tested. Spectra of the velocity product responsible for the mean shear stress and the lateral gradient of this spectrum have been determined, as has a structure constant for the lateral velocity fluctuations that Phillips (1967) suggested is relevant to the maintenance of the shear stress gradient. Phillips’ (1967) stress model fails the tests proposed in this study.

An experimental study of a plane mixing layer.

Examination of existing literature is revealing in that although many investigations have been carried out for a plane mixing layer, only a few present turbulence measurements. It is well-known that both mean velocity and turbulence structure in a plane mixing layer are self-preserving, however there appears to be some variations in these measurements. This investigation presents new data on the mean velocity and the turbulence properties of a plane mixing layer. The turbulence measurements differ by about 25% from those obtained by previous investigators and this is attributed to differences in the experimental set up (i.e., presence or absence of a solid wall in the plane x = 0) and errors associated with hot-wire probes (i.e., longitudinal cooling and wake interference). To avoid difficulties experienced by previous investigators in calculating shear stress distribution from a measured nondimensional mean velocity profile a new method is suggested and this provides good agreement between the calculated and the measured shear stress distribution. ECENTLY, considerable effort has been directed towards the study of free turbulent shear flows. In this group of shear flows a plane, turbulent, incompressible mixing layer between a uniform stream and quiescent surroundings is a com- paratively simple flow to investigate both theoretically and experimentally. However, except for a few investigations (as for example Garshore1; Hackett and Cox2) dealing with mean velocity measurements not much renewed attention appears to have been given to the plane turbulent mixing layer since the appearance of the work of Liepmann and Laufer.3 It should be recalled that Liepmann and Laufer did not measure both the lateral turbulent intensity and the shear stress presumably because is expected to be of the same magnitude as and is expected to be zero. Furthermore, techniques of hot-wire anemometry have developed considerably since their investigation and therefore it is of interest to reinvestigate the plane mixing layer. Since the completion of the present investigation Wygnanski and Fiedler4 have reported extensive and sophisticated turbu- lence measurements in this type of flow. An interesting aspect of their investigation was the geometry of the experimental apparatus. They used a trip wire and a solid wall on the zero velocity side in the plane x = 0. They also used conventional DISA x-wire probes to measure the turbulence parameters and it is now known that the conventional DISA x-wire probes are contaminated by the thermal cross talk.5 The measurements of Wygnanski and Fiedler are compared with the results of the present investigation obtained with single hot-wires in a plane mixing layer without a solid wall in the plane x = 0. It is anticipated that such a comparison would show up differences, if any, due to the geometry of the experimental apparatus and the probes. Recently Spencer and Jones6 have investigated a general problem of a mixing layer between two parallel streams and as a special case they have reported some data on a plane mixing layer. This information consists of the spreading rate and the mean velocity profile. Although not directly related to the

The two-dimensional mixing region

The two-dimensional incompressible mixing layer was investigated by using constant-temperature, linearized hot wire anemometers. The measurements were divided into three categories: (1) the conventional average measurements; (2) time-average measurements in the turbulent and the non-turbulent zones; (3) ensemble average measurements conditioned to a specific location of the interface. The turbulent energy balance was constructed twice, once using the conventional results and again using the turbulent zone results. Some differences emerged between the two sets of results. It appears that the mixing region can be divided into two regions, one on the high velocity side which resembles the outer part of a wake and the other on the low velocity side which resembles a jet. The binding turbulent–non-turbulent interfaces seem to move independently of each other. There is a strong connexion between the instantaneous location of the interface and the axial velocity profile. Indeed the well known exponential mean velocity profile never actually exists at any given instant. In spite of the complexity of the flow the simple concepts of eddy viscosity and eddy diffusivity appear to be valid within the turbulent zone.

Planar Free Turbulent Mixing With an Axial Pressure Gradient

An analysis of incompressible free turbulent mixing processes considering either a velocity defect or excess and both favorable and adverse pressure gradients is presented. The full nonsimilar flow field is treated directly by employing a linearized approximation to the momentum equation in the von Mises plane with the total pressure as the dependent variable. A “universal” solution results which can be used to determine the velocity field for any particular pressure gradient by simple algebraic manipulation. Satisfactory agreement with experimental data is shown, and some parametric studies of pressure gradient effects are presented.

Erratum: “Incompressible Jet Mixing in Converging-Diverging Axisymmetric Ducts” (Journal of Basic Engineering, 1967, 89, pp. 210–220)

Erratum: “Incompressible Jet Mixing in Converging-Diverging Axisymmetric Ducts” (Journal of Basic Engineering, 1967, 89, pp. 210–220)

Incompressible Jet Mixing in Converging-Diverging Axisymmetric Ducts

A test has been made of the constant turbulent Reynolds number hypothesis for a turbulent jet flow in a converging-diverging axisymmetric tube. Using the free turbulent jet as the sole source of data for evaluation of the Reynolds number, and velocity shear distributions, the general ducted flow was predicted with the aid of appropriate similarity relations. Calculated results are compared with the extensive and consistent data of Helmbold; good agreement is observed. It is found that the methods of calculation employed can be considerably simplified without large effect on the calculated results. The existence and extent of zones of recirculation are also discussed.