Streamline Coordinate System Research Articles

The vortex dynamics characteristics in a pump-turbine: A rigid vorticity analysis while varying guide vane openings in turbine mode

In off-design conditions, pump-turbines generate vortex structures of varying scales and intensities, jeopardizing energy conversion due to unstable flow phenomena. While off-design pump-turbine internal flow dynamics have been analyzed, understanding the spatiotemporal evolution of complex vortex structures within runner blade passages, which are prone to inducing hydraulic instabilities, has to be still addressed properly. This paper applies the rigid vorticity theory, also known as Liutex, to explore vortex characteristics in the pump-turbine under different guide vane openings during turbine mode operation. The investigation focuses on three main aspects: i) spatial distribution characteristics, ii) temporal evolution characteristics, and iii) vortex morphological features along with deformation mechanisms. Results showed that the suction surface side of the blade's trailing edge commonly experiences significant shear effects, while vortices tend to emerge on both pressure and suction surface sides. Under small Guide Vane Opening (GVO) condition, vortices in blade passages show periodic evolution, and Enstrophy of the Rigid Vorticity Dilatation Term (ERDT) predominantly contributes to the helical deformation of these vortices. Conversely, under large GVO condition, vortices maintain high helicity and remain stable. This study provides insights to reduce hydraulic instability in pump-turbines and optimize energy conversion efficiency. The proposed Reynolds-averaged transport equation for enstrophy of the rigid vorticity and the relative streamline coordinate system provide advanced tools for analyzing unstable flow structures.

Fluid deformation in isotropic Darcy flow

The deformation of fluid elements governs many processes in porous media, including the transport, dispersion and mixing of solutes, chemical reactions and colloidal particles. Recently, it has been shown that even in strongly heterogeneous porous media, steady three-dimensional (3-D) Darcy flow with isotropic hydraulic conductivity admits only constrained kinematics. This is due to its inherently helicity-free nature, as the velocity is everywhere orthogonal to the vorticity. This property precludes braided streamlines and instead admits a pair of coherent 3-D streamfunctions, and the streamlines cannot wander freely throughout the flow domain. In this study, we consider the impact of these kinematic constraints upon fluid deformation at the Darcy scale. We show that the helicity-free condition corresponds to an orthogonal 3-D streamline coordinate system, which we use to derive anab initiocontinuous time random walk framework for fluid deformation. We find that the helicity-free condition combined with the intermittent nature of shear events leads to fluid deformation that is limited to algebraic growth, with stretching ranging from sublinear to superlinear behaviour. Fluid deformation in 3-D isotropic Darcy flow is remarkably similar to that of two-dimensional (2-D) Darcy flow, and the structure of 3-D Darcy flow is fundamentally the same as two superposed 2-D Darcy flows. These results have implications for understanding flow and transport in heterogeneous porous media, and provide a basis for quantification of mixing and dispersion in isotropic 3-D Darcy flow.

Numerical investigation of secondary flows in a constant-width wind-tunnel contraction

AbstractThe flow inside a constant-width wind-tunnel contraction is simulated by solving the Reynolds-Averaged Navier-Stokes equations with an eddy-viscosity turbulence model. The results show the presence of longitudinal vortices near the sidewalls centreline. This confirms a former hypothesis involving the generation of skew-induced longitudinal vorticity within the sidewalls boundary layers. Detailed analysis reveals that the flow structure is influenced by viscous effects in the boundary layers and streamline curvature in the potential flow. Three-dimensional boundary-layer profiles on the contraction sidewall are analysed in the framework of the streamline co-ordinate system and its associated hodographic diagram. The resulting profiles help understand the generation of secondary flows and the associated longitudinal vorticity.

Estimating the mean flow field in combustion chambers

Swirling flow fields in combustion chambers can be determined based on swirl ratio and a velocity profile specified along some path to the vortex center. A method is presented whereby flow fields can be constructed by applying the continuity equation in a streamline coordinate system and imposing irrotationality about the symmetry axis of the vortex ring. The swirl ratio may be specified at the vortex core, along with a velocity profile along any semi-axis of the vortex cross section.

The influence of topography on single-tower-based carbon flux measurements under unstable conditions: a modeling perspective

Based on high-resolution numerical simulations, the influence of topography on the one-dimensional (1-D) tower-based measurements of the net ecosystem–atmosphere exchange rate (NEE) of CO2 was analyzed under unstable conditions. Airflow and transport of a passive scalar were simulated over undulating surfaces covered by tall trees. Compared to their flat surface counterparts, the wind and scalar mixing ratio fields are more disturbed over a steeper surface and/or under a weaker background wind condition, resulting in larger errors in the 1-D NEE estimation. The magnitude of the error is generally larger on the windward side than on the lee side when topography-induced circulation (TIC) does not occur. Applying the ensemble streamline coordinate system to estimating NEE could result in larger errors than applying the local earth coordinate system or terrain coordinate system when the atmospheric flow is significantly distorted. This is especially true when TIC occurs. NEE estimated from the 1-D framework under convective conditions with calm or weak background winds may incur significant errors even over gentle topography. Preference in the selection of a flux tower location is given to the crest area.

Numerical solution of three‐dimensional viscous shock layer using axisymmetric analog along the streamlines

PurposeThis paper aims to predict aerodynamic heating through the efficient solution of three‐dimensional viscous shock layer (VSL) equations, using axisymmetric analog.Design/methodology/approachThe three‐dimensional VSL equations are written in the curvilinear streamline coordinate system. In these equations, normal momentum equation is replaced by Maslen's pressure relation. In addition to this, axisymmetric analog is implemented along the streamlines through assuming a zero value for circumferential velocity component. In this case, three‐dimensional VSL equations are reduced into an axisymmetric form, which can be solved much easier.FindingsIt is demonstrated that the solution of three‐dimensional VSL equations in the curvilinear streamline coordinate system, using axisymmetric analog, has made it possible to predict convective heat fluxes in both windward and leeward regions. Moreover, in comparison with the 3D VSL methods, the present approach dramatically reduces the CPU time of calculations. Comparison with the experimental and numerical data shows a good agreement between both of these data and the present results.Practical implicationsThis method is an excellent tool for parametric study and preliminary design of hypersonic vehicles.Originality/valueThis method can predict convective heat flux in the leeward region where other similar methods are not applicable. In addition to this the present method is faster than other methods of solution for the 3D VSL equations.

Analytical Study of Turbulent Pollutant Dispersion near a Low Hill

An analytical methodology is developed to study the pollutant dispersion in a turbulent wind flow over a two-dimensional hill with a small slope. As in a typical boundary layer problem, the flow domain is divided into an inner and an outer region: the inviscid outer region is further subdivided into an upper and a middle layer while the viscous inner region is subdivided into a shear stress and an inner surface layer. Based on the Reynolds-averaged Navier-Stokes equations and the continuity equations, closed form analytical solutions of the stream functions and velocities are readily obtained for all regions in the domain. The velocity information is then imported into the diffusion equation, and the pollutant concentration distribution is readily solved. For reasons of turbulent shear, a variational method with adjustments to the streamline coordinate system is used to obtain an accurate solution of the pollutant concentration. Results show that when the source is located in the upper layer, the concentrations decrease with distance along the upwind side of the hill and tend to reach a constant value rapidly near the hilltop. Similar results are observed when the source is located in the middle layer. However, due to the reduction of wind speed in the middle layer, the concentrations become saturated at a later upslope position as compared to the source in the upper layer. This methodology is shown to be able to provide a quick and accurate estimate of local pollutant patterns and can be applied to any flow field provided that the streamlines can be specified through the velocities.

Lateral dynamic analysis and classification of barotropic tidal inlets

The dynamical balances at shallow tidal inlets are highly nonlinear, and can vary substantially over sub-kilometer scales. In this study, barotropic dynamics are examined with numerical experiments on a series of idealized inlets with differing inlet widths and lengths. Circulation and elevation fields obtained from fully nonlinear depth-integrated circulation models are used to reconstruct the contribution of each term in the momentum equations. Momentum terms are rotated into a streamline coordinate system to simplify interpretation of the dynamics. Spatial patterns in momentum reveal that the lateral balances at inlets can vary from nearly geostrophic to strongly cyclostrophic. Marked dynamical differences are seen between inlets with different lengths and widths. Inlet regions of geostrophic or cyclostrophic balances can be predicted using two dimensionless parameters, the dynamic length L * and dynamic width W *. A classification scheme is proposed using L * and W * to compare the idealized inlets analyzed here with inlets from 20 previous studies. Four distinct inlet types are identified and discussed.

Electroosmotic flows in microchannels with finite inertial and pressure forces.

Emerging microfluidic systems have spurred an interest in the study of electrokinetic flow phenomena in complex geometries and a variety of flow conditions. This paper presents an analysis of the effects of fluid inertia and pressure on the velocity and vorticity field of electroosmotic flows. In typical on-chip electrokinetics applications, the flow field can be separated into an inner flow region dominated by viscous and electrostatic forces and an outer flow region dominated by inertial and pressure forces. These two regions are separated by a slip velocity condition determined by the Helmholtz-Smoulochowski equation. The validity of this assumption is investigated by analyzing the velocity field in a pressure-driven, two-dimensional flow channel with an impulsively started electric field. The regime for which the inner/outer flow model is valid is described in terms of nondimensional parameters derived from this example problem. Next, the inertial forces, surface conditions, and pressure-gradient conditions for a full-field similarity between the electric and velocity fields in electroosmotic flows are discussed. A sufficient set of conditions for this similarity to hold in arbitrarily shaped, insulating wall microchannels is the following: uniform surface charge, low Reynolds number, low Reynolds and Strouhal number product, uniform fluid properties, and zero pressure differences between inlets and outlets. Last, simple relations describing the generation of vorticity in electroosmotic flow are derived using a wall-local, streamline coordinate system.

Sonic Anemometer Tilt Correction Algorithms

The sensitivity of sonic anemometer-derived stress estimates to the tilt of the anemometer is investigated. The largest stress errors are shown to occur for unstable stratification (z/L<0) and deep convective boundary layers. Three methods for determining the tilt angles relative to a mean streamline coordinate system and for computing the tilt-corrected stresses are then compared. The most commonly used method, involving a double rotation of the anemometers' axes, is shown to result in significant run-to-run stress errors due to the sampling uncertainty of the mean vertical velocity. An alternative method, requiring a triple rotation of the anemometer axes, is shown to result in even greater run-to-run stress errors due to the combined sampling errors of the mean vertical velocity and the cross-wind stress. For measurements over the sea where the cross-stream stress is important, the double rotation method is shown to overestimate the surface stress, due to the uncorrected lateral tilt component. A third method, using a planar fit technique, isshown to reduce the run-to-run stress errors due to sampling effects, and provides an unbiased estimate of the lateral stress.

Numerical Analysis of the Flow in a Poppet Valve Using a Streamline Coordinate System

Numerical Analysis of the Flow in a Poppet Valve Using a Streamline Coordinate System

Numerical Analysis of a Jet Using a Streamline Coordinate System. Annular Jet Issuing from a Coaxial Pipe.

This paper describes the analysis of a jet using a streamline coordinate system. Up to now, the (streamline function) ω (vorticity) method has been used for such a coordinate system. By this method, however, pressure does not appear as an explicit variable, so that some technique is needed to satisfy the pressure boundary conditions. To overcome this problem, the SIMPLER method is used to solve the governing equations in the streamline coordinates, because in this way the pressure boundary conditions can be satisfied. To show the usefulness of this method, the annular jet from a coaxial pipe was taken as an example. While previous studies on this subject assumed that the velocity at any section of the jet was uniform, we took into account the velocity distribution. Our computed results were in good agreement with the experimental ones in Ref. 8, and so, the usefulness of the method was confirmed.

Simple Numerical Method to Compute Viscous Lift Loss of Wings

It is well known that the lift of a wing is less than the value computed on the basis of potential  ow because of the presence of a boundary layer on the surface. A method of computing this lift loss is described based on numerically solving the integral equations of the three-dimensional boundary-layer  ow in a streamline coordinate system. When applied to several swept, tapered, and twisted wings, the method closely predicts the lift curve measured in a wind tunnel. Similar computations with a twodimensional (strip) model overestimate wing viscous lift loss by about 10%. The numerical effort required to solve the integral boundary-layer equation set is much less than that required to solve either the threedimensional boundary-layer equations or the Navier – Stokes equations, with little loss in accuracy. It could easily be used, therefore, for preliminary estimates of climb and cruise wing performance. A simple procedure is presented for identifying cross ow separation.

A Method of Analysis for Turbulent Flows Using the Streamline Coordinate System

A procedure is described for analysing the transport equations for Reynolds stresses written in a streamline coordinate system, starting from the fields of first- and second-order moments of wind velocity measured in a terrain-following system over topography. In the analysis, the equations are split into two parts: the first contains the terms that can be calculated directly from measurements; the second contains third-order moments that are parameterized using suitable models. To evaluate the error associated with both parts, a Monte-Carlo technique that takes into account the experimental errors is proposed. An example of the application of this method for the Reynolds shear stress equation, using wind-tunnel data for non-separating flow over a two-dimensional valley, is reported. The comparison between the measured and modelled parts is fair near the surface, while at higher levels, the modelled part can be shown to miss a correct treatment of the third-order moments. In the frame of this analysis, the use of the correct derivative transformation has been found to be significant even for moderately sloping topography.

A diagnostic study of turbulence features in wind tunnel flow over simple topographic shapes

The turbulence features observed in wind tunnel experiments of flow over isolated simple-shaped two-dimensional hills and valleys are analysed in the light of transport equations of Reynolds stresses in the streamline coordinate system. The analysis has shown that near the surface the streamwise pressure gradient is the driving parameter for all the stresses. At higher levels, the effects of streamline curvature and streamwise pressure gradient act differently on the source terms of stresses, showing that the shear stress is the most sensitive to streamline curvature, whereas the streamwise stress is the most sensitive to streamwise pressure gradient. Although two of the considered topographies are specular images, the turbulence response is not so, since it depends on the sequence of curvatures to which it is subjected. The information and analysis presented will constitute a valuable resource to those seeking an understanding of atmospheric boundary layer development over complex terrain. It will also provide a support for the development and validation of more rigorous turbulence models.

Dispersion of Tracers by Ocean Gyres

Abstract The dispersion of a tracer by a two-dimensional gyre circulation is studied using simple numerical models. Two approaches are taken: a random walk model formulated in a streamline coordinate system and the numerical solution of the advection-diffusion equation. A number of different gyres are considered. Attention is focused on the characteristics of the gyre that determine the spreading and mixing time of the tracer. The authors find that the dispersion by a given gyre can be characterized in terms of a bulk Peclet number and the three length scales: L the horizontal width of the gyre, l the width of the boundary current, and L the length of the boundary current. By taking into account the length of the boundary layer, gyre dispersion is found to conform moderately well with previous analytic models, in particular the partitioning between weak and strong diffusive regimes, even though the shear characteristics may be quite variable across the gyre. The analytic models become less valid as the le...

Turbulent dispersion of pollutant over complex terrain

A two-dimensional turbulent diffusion equation is derived in a streamline coordinate system, defined for rotational flow over complex terrain and limited aloft by an elevated, impenetrable inversion. In the first instance, the steady-state equation is solved for an “inner region” of the boundary layer, in which the effect of curvature is negligible and, for simplicity, it is assumed that vorticity has a power-law dependence upon stream function. A variational method of solution is also discussed, in which vorticity may have a more general representation. A numerical calculation is performed for a special case of symmetrical flow over an isolated hill. The dependence of pollutant concentration upon the flow field, downwind distance and source is examined and the effect of wind acceleration in the neighbourhood of the top of the hill is discussed. It is pointed out that the diffusion model can be applied to any realistic flow field, provided that the streamlines are specified.

The structure of a three-dimensional turbulent boundary layer

The three-dimensional turbulent boundary layer is shown to have a self-consistent two-layer asymptotic structure in the limit of large Reynolds number. In a streamline coordinate system, the streamwise velocity distribution is similar to that in two-dimensional flows, having a defect-function form in the outer layer which is adjusted to zero at the wall through an inner wall layer. An asymptotic expansion accurate to two orders is required for the cross-stream velocity which is shown to exhibit a logarithmic form in the overlap region. The inner wall-layer flow is collateral to leading order but the influence of the pressure gradient, at large but finite Reynolds numbers, is not negligible and can cause substantial skewing of the velocity profile near the wall. Conditions under which the boundary layer achieves self-similarity and the governing set of ordinary differential equations for the outer layer are derived. The calculated solution of these equations is matched asymptotically to an inner wall-layer solution and the composite profiles so formed describe the flow throughout the entire boundary layer. The effects of Reynolds number and cross-stream pressure gradient on the cross-stream velocity profile are discussed and it is shown that the location of the maximum cross-stream velocity is within the overlap region.

Calculation of Three-Dimensional Boundary Layers on Rotor Blades Using Integral Methods

The important effects of rotation and compressibility on rotor blade boundary layers are theoretically investigated. The calculations are based on the momentum integral method and results from calculations of a transonic compressor rotor are presented. Influence of rotation is shown by comparing the incompressible rotating flow with the stationary one. Influence of compressibility is shown by comparing the compressible rotating flow with the incompressible rotating one. Two computer codes for three-dimensional laminar and turbulent boundary layers, originally developed by SSPA Maritime Consulting AB, have been further developed by introducing rotation and compressibility terms into the boundary layer equations. The effect of rotation and compressibility on the transition have been studied. The Coriolis and centrifugal forces that contribute to the development of the boundary layers and influence its behavior generate crosswise flow inside the blade boundary layers, the magnitude of which depends upon the angular velocity of the rotor and the rotor geometry. The calculations show the influence of rotation and compressibility on the boundary layer parameters. Momentum thickness and shape factor increase with increasing rotation and decrease when compressible flow is taken into account. For skin friction such effects have inverse influences. The different boundary layer parameters behave similarly on the suction and pressure sides with the exception of the crossflow angle, the crosswise momentum thickness, and the skin friction factor. The codes use a nearly orthogonal streamline coordinate system, which is fixed to the blade surface and rotates with the blade.

Flow in a Poppet Valve. Computation of Pressure Distribution using a Streamline Coordinate System.

The steady axisymmetric laminar flow in a poppet valve is analyzed by means of a conventional finite difference method (ψ-ω method) . After obtaining the numerical results for stream functionψ and vorticity ω in the cylindrical coordinate system, a streamline coordinate system (φ, ψ) is applied to compute pressure distributions, because good pressure distributions cannot be obtained using the cylindrical coordinates. The pressure on the valve body surface is experimentally measured. The numerical results for pressure are in good agreement with the experimental ones.