Accuracy of tomographic particle image velocimetry data on a turbulent round jet

Tomographic particle image velocimetry (Tomo-PIV) was applied on a turbulent round air jet to quantitatively assess the accuracy of velocity gradients obtained in the self-similar turbulent region. The jet Reynolds number based on the nozzle diameter (d) was Re_d = 3000. Mean velocity, turbulent intensities, and Reynolds shear stress at the center plane of the jet were measured. In addition, statistical results of Tomo-PIV along the axial direction were assessed by performing a separate set of two-dimensional two-component PIV experiments on a “side view” plane along the jet axis. Moreover, the probability distribution functions of four components of the measured velocity gradients in the axial and radial directions were validated by these “side view” planar PIV data. The root mean square of the velocity divergence values relative to the norm of the velocity gradient tensor was 0.36. Furthermore, the on- and off-diagonal components of the velocity gradients satisfied the axisymmetric isotropy conditions. The divergence error in the data affected only areas with low gradient magnitude. Therefore, turbulent structures in the regions with intense vorticity and dissipation can be closely monitored. On this basis, the joint pdfs of the invariants of the velocity gradient and strain and rotation tensor rates were produced and compared well with those in isotropic turbulence studies.     

SEMI-IMPLICIT FINITE VOLUME SHALLOW-WATER FLOW AND SOLUTE TRANSPORT SOLVER WITH k–ε TURBULENCE MODEL

A 3D semi-implicit finite volume scheme for shallow- water flow with the hydrostatic pressure assumption has been developed using the σ-co-ordinate system, incorporating a standard k–ε turbulence transport model and variable density solute transport with the Boussinesq approximation for the resulting horizontal pressure gradients. The mesh spacing in the vertical direction varies parabolically to give fine resolution near the bed and free surface to resolve high gradients of velocity, k and ε. In this study, wall functions are used at the bed (defined by the bed roughness) and wind stress at the surface is not considered. Surface elevation gradient terms and vertical diffusion terms are handled implicitly and horizontal diffusion and source terms explicitly, including the Boussinesq pressure gradient term due to the horizontal density gradient. The advection terms are handled in explicit (conservative) form using linear upwind interpolation giving second-order accuracy. A fully coupled solution for the flow field is obtained by substi- tuting for velocity in the depth-integrated continuity equation and solving for surface elevation using a conjugate gradient equation solver. Evaluation of horizontal gradients in the σ-co-ordinate system requires high-order derivatives which can cause spurious flows and this is avoided by obtaining these gradients in real space. In this paper the method is applied to parallel oscillatory (tidal) flow in deep and shallow water and compared with field measurements. It is then applied to current flow about a conical island of small side slope where vortex shedding occurs and velocities are compared with data from the laboratory. Computed concentration distributions are also compared with dye visualization and an example of the influence of temperature on plume dispersion is presented. © 1997 John Wiley & Sons, Ltd.     

The reduction of skin friction by riblets under the influence of an adverse pressure gradient

In this paper we address the effectiveness of riblets on skin friction reduction under the influence of an adverse pressure gradient. The measurements were taken in a wind tunnel. Skin friction was observed with a drag balance which has a reproducibility of better than 1%. The accuracy of the balance is estimated to be less than 1% for the case of zero-pressure gradient and at most 3% for a pressure gradient. The data on skin friction reduction at zero pressure gradient were consistent with previous results and amount to 5% at dimensionless riblet width of s ⁺ = 13. We find that at all adverse pressure gradients the skin friction reduction by riblets persists. At moderate pressure gradients the reduction increases somewhat to 7%. The velocity profile which is also measured, exhibits the characteristic shape for a boundary layer with an adverse pressure gradient and agrees well with theory. From the velocity profiles measured at two stations we estimated with the help of a momentum balance the skin friction and skin friction reduction. The results differ from the drag-balance data. Due to the poor accuracy of the momentum balance method which we estimate in our case, we conclude that the results obtained with this method are less reliable than those obtained with the drag balance. This throws some doubt on previous results on drag reduction under the influence of a pressure gradient which were based on the momentum balance method.     

Amplitude modulation and its relation to streamwise convection velocity

The paper is devoted to show the relation between the amplitude modulation and the convection velocity in a turbulent boundary layer. The analysis has been performed based upon velocity profiles measured with the hot-wire technique in a turbulent boundary layer with pressure gradient corresponding to turbomachinery conditions. The VITA detection method of small-scale structures was used in order to show the impact of the pressure gradient on phase averaged time-traces changes in the aspect of amplitude modulation. The physical significance of the large scales influence on the small scales, formerly known as the amplitude modulation, has been elucidated. It was demonstrated that the measure of amplitude modulation can be used as an indicator of the convection velocity of coherent structures. The convection velocity profile can be described in the inner region using the amplitude modulation term when the proper scale is applied. The convection velocity prediction in favourable and adverse pressure gradients shows that t the strongest impact of the pressure gradient occurs in the buffer layer. The stronger increase occurs in the buffer layer. It was also found that the skewness factor of phase-averaged VITA waveforms, is very similar in shape with the cross-product term of the streamwise skewness factor, which quantifies amplitude modulation. It indicates that the well-known modification of quadrant events in pressure gradient flows is a result of the amplitude modulation phenomenon.     

On Fourier's law of heat conduction

A linear theory of fluid is considered in which the gradients of density, internal energy and velocity are among the constitutive variables. Thus the heat flux may be a linear combination of the gradients of density and internal energy. It is proved that this linear combination may be written as the gradient of temperature so that Fourier's law of heat conduction holds.     

The measurement of velocity gradients in fluid flow by laser light scattering

The motion of the speckle pattern in the far field limit is uniquely determined by the velocity gradient of the light scattering medium. This is the basic physical effect of a non-invasive method that allows the measurement of velocity gradients with high spatial and temporal resolution down to 75 m and 1 ms, respectively. We present three different measurement schemes for the analysis of the scattered light. The experimental results refer to mean gradients obtained from measurements on turbulent channel flows.     

Optimization of multi-stage velocity gradients in a cylindrical fluidized bed flocculator

Flocculation time is conventionally believed to be proportional to the flocculation efficiency of a cylindrical fluidized bed flocculator. However, in a single-stage velocity gradient situation, the flocculation efficiency decreases when the optimal flocculation time is exceeded. A multi-stage velocity gradient was established in a cylindrical fluidized bed flocculator, based on the hydraulic classification theory. This multi-stage velocity gradient fluidized bed flocculator (MGF) created a more suitable environment for floc growth and protection, which was confirmed by the size distribution of flocs along the bed height. Correspondingly, the abatement efficiencies for Kaolin slurry and dyed wastewater treatment in the MGF were enhanced by 5–10%, and by 7–20%, respectively, compared with those in the single-stage velocity gradient fluidized bed flocculators (SGFs). The initial bed height distribution ratio along the velocity gradients was an important factor for MGF optimization.     

基于分形理论的煤层气非达西流动分析

在考虑了煤层的分形特征和启动压力梯度影响的基础上,建立了无限大煤层中气体低速非达西流动的数学模型,并求得了量纲为一的井底压力的Laplace空间解析解,并根据数值求解结果绘制了典型的井底压力动态曲线。研究结果表明:在定产量生产的情况下,分形维数和量纲为一的启动压力梯度对早期井底压力动态无显著影响;在生产的中后期,由于受二者的影响,压力导数曲线上的径向流水平直线段消失;量纲为一的井储系数的影响主要表现在续流阶段,而吸附系数则主要影响煤基质向裂隙扩散的过渡阶段。     

Constitutive equations for multiphase mixtures of fluids

Constitutive equations for a multiphase mixture of fluids are presented. The mixture is assumed to consist of a single non-uniform temperature and no change is allowed. The theory is based on the conservation and balance equations of multiphase mixtures proposed by Dobran, and the constitutive assumption allows for the effects of temperature gradient, density gradients, velocity gradients, velocities and accelerations. A linearized form of the constitutive equations is presented for an arbitrary number of phases, and restrictions on the constitutive assumption are investigated by the second law of thermodynamics. The theory yielded a significant number of results and they are compared with previous investigations.     

Determination of complete velocity gradient tensor by using cinematographic stereoscopic PIV in a turbulent jet

Cinematographic stereoscopic PIV measurements were performed in the far field of an axisymmetric co-flowing turbulent round jet (Re _T ≈ 150, where Re _T is the Reynolds number based on Taylor micro scale) to resolve small and intermediate scales of turbulence. The time-resolved three-component PIV measurements were performed in a plane normal to the axis of the jet and the data were converted to quasi-instantaneous three-dimensional (volumetric) data by using Taylor’s hypothesis. The availability of the quasi-three-dimensional data enabled the computation of all nine components of the velocity gradient tensor over a volume. The use of Taylor’s hypothesis was validated by performing a separate set of time-resolved two component “side-view” PIV measurements in a plane along the jet axis. Probability density distributions of the velocity gradients computed using Taylor’s hypothesis show good agreement with those computed directly with the spatially resolved data. The overall spatial structure of the gradients computed directly exhibits excellent similarity with that computed using Taylor’s hypothesis. The accuracy of the velocity gradients computed from the pseudo-volume was assessed by computing the divergence error in the flow field. The root mean square (rms) of the divergence error relative to the magnitude of the velocity gradient tensor was found to be 0.25, which is consistent with results based on other gradient measurement techniques. The velocity gradients, vorticity components and mean dissipation in the self-similar far field of the jet were found to satisfy the axisymmetric isotropy conditions. The divergence error present in the data is attributed to the intrinsic uncertainty associated with performing stereoscopic PIV measurements and not to the use of Taylor’s hypothesis. The divergence error in the data is found to affect areas of low gradient values and manifests as nonphysical values for quantities like the normalized eigenvalues of the strain-rate tensor. However, the high gradients are less affected by the divergence error and so it can be inferred that structural features of regions of intense vorticity and dissipation will be faithfully rendered.     

Diffusion and thermal slip of a binary gas mixture

Let us note that the phenomenon of diffusion slip at a constant gas-mixture temperature has been considered in [1], for example, and thermal slip for a single-component gas in [2]. The slip velocity of a binary gas mixture has been calculated in a field of the temperature gradient and of the partial pressure gradients. The kinetic equation is solved by an approximate method based on physical considerations. A formula has been obtained analytically for the slip velocity for arbitrary accommodation coefficients as well as for arbitrary gas concentrations and arbitrary molecule masses. The results agree to 1% accuracy with the numerical computations of other authors.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 51–55, July–August, 1970.     

On the description of the motion of the yield surface in unsteady shearing flows of a Bingham fluid as a jerk wave

In the unsteady shearing flows of a Bingham fluid, the yield surface may move laterally into the fluid with a finite speed. The exact nature of this motion can be explained by assuming that it is a jerk wave. That is, the yield surface is a singular surface across which the velocity, the acceleration and the velocity gradient are continuous, whereas the jerk, which is the time derivative of the acceleration, the spatial gradient of the acceleration and the second gradient of the velocity all suffer jumps. Simultaneously, across this singular surface, the shear stress, its time derivative and its gradient are continuous, while the corresponding temporal and spatial gradients of second order suffer jumps, with Hadamard’s Lemma defining the speed of propagation of the jerk wave. These theoretical assumptions are found to hold true in a shearing flow of a Bingham fluid in an unbounded domain, studied by Sekimoto. It is further shown that the same kinematical and dynamical conditions explain the movement of yield surfaces in the shearing flows of all viscoplastic fluids.     

Spatial representation of evolving anisotropy at large strains

A phenomenological model for evolving anisotropy at large strains is presented. The model is formulated using spatial quantities and the anisotropic properties of the material is modeled by including structural variables. Evolution of anisotropy is accounted for by introducing substructural deformation gradients which are linear maps similar to the usual deformation gradient. The evolution of the substructural deformation gradients is governed by the substructural plastic velocity gradients in a manner similar to that for the continuum. Certain topics related to the numerical implementation are discussed and a simple integration scheme for the local constitutive equations is developed. To demonstrate the capabilities of the model it is implemented into a finite element code. Two numerical examples are considered: deformation of uniform plate with circular hole and the drawing of a cup. In the two examples it is assumed that initial cubic material symmetry applies to both the elastic and plastic behavior. To be specific, a polyconvex Helmholtz free energy function together with a yield function of quadratic type is adopted.     

Turbulent boundary layer velocity profiles over a smooth surface

Mean velocity measurements made for two-dimensional incompressible turbulent boundary layers, for 3 pressure gradients on a smooth wall have been used to evaluate the pressure gradient parameter, the skin-friction and the roughness function by the method of “curve-fit” to the entire profile. Local skin-friction coefficients so obtained do not agree with the experimental results but the integral parameters are well represented.     

Improvements to PIV image analysis by recognizing the velocity gradients

Two iterative PIV image processing methods are introduced, which utilize displacement and deformation of the interrogation areas to maximize the correlation. The velocity gradients used for the window deformation are iteratively estimated directly from the images and no velocity values are required from neighbouring interrogation areas, as with numerical differentiation. The improved accuracy and resolution of the velocity gradient estimation compared to numerical differentiation is shown using synthetic images. The performance in a real application is shown using experimental reference images.     

A parametric study of adverse pressure gradient turbulent boundary layers

There are many open questions regarding the behaviour of turbulent boundary layers subjected to pressure gradients and this is confounded by the large parameter space that may affect these flows. While there have been many valuable investigations conducted within this parameter space, there are still insufficient data to attempt to reduce this parameter space. Here, we consider a parametric study of adverse pressure gradient turbulent boundary layers where we restrict our attention to the pressure gradient parameter, β, the Reynolds number and the acceleration parameter, K. The statistics analyzed are limited to the streamwise fluctuating velocity. The data show that the mean velocity profile in strong pressure gradient boundary layers does not conform to the classical logarithmic law. Moreover, there appears to be no measurable logarithmic region in these cases. It is also found that the large-scale motions scaling with outer variables are energised by the pressure gradient. These increasingly strong large-scale motions are found to be the dominant contributor to the increase in turbulence intensity (scaled with friction velocity) with increasing pressure gradient across the boundary layer.     

A novel architecture for a super-resolution PIV algorithm developed for the improvement of the resolution of large velocity gradient measurements

Three different particle image processing algorithms have been developed for the improvement of PIV velocity measurements characterized by large velocity gradients. The objectives of this study are to point out the limitations of the standard processing methods and to propose a complete algorithm to enhance the measurement accuracy. The heart of the PIV image processing is a direct cross-correlation calculation in order to obtain complete flexibility in the choice of the size and the shape of the interrogation window (IW). An iterative procedure is then applied for the reduction of the size of IW at each measurement location. This procedure allows taking into account the local particle concentration in the image. The results of this first iterative processing, applied to synthetic images, show both a significant improvement of measurement accuracy and an increase of the spatial resolution. Finally, a super-resolution algorithm is developed to further increase the spatial resolution of the measurement by determining the displacement of each particle. The computer time for a complete image processing is optimized by the introduction of original data storage in Binary Space Partitions trees. It is shown that measurement errors for large velocity gradient flows are similar to those obtained in simpler cases with uniform translation displacements. This last result validates the ability of the developed super-resolution algorithm for the aerodynamic characterization of large velocity gradient flows.     

Investigation of internal pressure gradients generated in electrokinetic flows with axial conductivity gradients

Field amplified sample stacking (FASS) is used to increase sample concentrations in electrokinetic flows. The technique uses conductivity gradients to establish a non-uniform electric field that accumulates ions within a conductivity gradient, and can be readily integrated with capillary electrophoresis. Conductivity gradients also cause gradients in near-wall electroosmotic flow velocities. These velocity gradients generate internal pressure gradients that drive secondary, dispersive flows. This dispersion leads to a significant reduction in the efficiency of sample stacking. This paper presents an experimental investigation of internally generated pressure gradients in FASS using micron-resolution particle image velocimetry (μPIV). We measure velocity fields of particles seeded into an electrokinetic FASS flow field in a glass microchannel with a single buffer–buffer interface. μPIV allows for the direct quantification of local, instantaneous pressure gradients by analyzing the curvature of velocity profiles. Measurements show internally generated pressure-driven velocities on the order of 1mm/s for a typical applied electric field of 100 V/cm and a conductivity ratio of 10. A one-dimensional (1D) analytical model for the temporal development of the internal pressure gradient generation is proposed which is useful in estimating general trends in flow dynamics.

---

Interfacial PIV to resolve flows in the vicinity of curved surfaces

PIV measurements near a wall are generally difficult due to low seeding density, low velocity, high velocity gradient, and strong reflections. Such problems are often compounded by curved boundaries, which are commonly found in many industrial and medical applications. To systematically solve these problems, this paper presents two novel techniques for near-wall measurement, together named Interfacial PIV, which extracts both wall-shear gradient and near-wall tangential velocity profiles at one-pixel resolution. To deal with curved walls, image strips at a curved wall are stretched into rectangles by means of conformal transformation. To extract the maximal spatial information on the near-wall tangential velocity field, a novel 1D correlation function is performed on each horizontal pixel line of the transformed image template to form a “correlation stack”. This 1D correlation function requires that the wall-normal displacement component of the particles be smaller than the particle image diameter in order to produce a correlation signal. Within the image regions satisfying this condition, the correlation function yields peaks that form a tangential velocity profile. To determine this profile robustly, we propose to integrate gradients of tangential velocity outward from the wall, wherein the gradient at each wall-normal position is measured by fitting a straight line to the correlation peaks. The capability of Interfacial PIV was validated against Particle Image Distortion using synthetic image pairs generated from a DNS velocity field over a sinusoidal bed. Different velocity measurement schemes performed on the same correlation stacks were also demonstrated. The results suggest that Interfacial PIV using line fitting and gradient integration provides the best accuracy of all cases in the measurements of velocity gradient and velocity profile near wall surfaces.