Case Of Unsteady Flow Research Articles

A solver using Newton's method for unsteady viscous flows

A solver is developed for time-accurate computations of viscous flows based on the conception of Newton's method. A set of pseudo-time derivatives are added into governing equations and the discretized system is solved using GMRES algorithm. Due to some special properties of GMRES algorithm, the solution procedure for unsteady flows could be regarded as a kind of Newton iteration. The physical-time derivatives of governing equations are discretized using two different approaches, i.e., 3-point Euler backward, and Crank-Nicolson formulas, both with 2nd-order accuracy in time but with different truncation errors. The turbulent eddy viscosity is calculated by using a version of Spalart-Allmaras one-equation model modified by authors for turbulent flows. Two cases of unsteady viscous flow are investigated to validate and assess the solver, i.e., low Reynolds number flow around a row of cylinders and transonic bi-circular-arc airfoil flow featuring the vortex shedding and shock buffeting problems, respectively. Meanwhile, comparisons between the two schemes of time-derivative discretizations are carefully made. It is illustrated that the developed unsteady flow solver shows a considerable efficiency and the Crank-Nicolson scheme gives better results compared with Euler method.

HYDRODYNAMIC FORCE EXERTING ON A SQURE PILLAR WITH ANGLE OF ATTACK BY A DAM BREAK FLOW

Hydrodynamic forces exerting on a squire pillar (length d) with angle of attack α, placed in the unsteady free surface shear flow, are investigated experimentally, being varied in the range of α=0°-45°. It is found that the assumption of hydrostatic pressure distribution at the front and back side of the pillar is not valid for the case of unsteady free surface shear flow and hydrodynamic force changes with the pressure difference Δh between the front and back side of the pillar. It is shown that Δh/d is dependent on the gradient dh/dx of water surface calculated from Ritter solution and Δh/d increases with |dh/dx|, and the drag coefficient Cd is smaller than that in uniform flows.

DNS and LES of Passively Controlled Turbulent Backward-Facing Step Flow

A passive control approach (no external energy input) for an unsteady separated flow case was investigated numerically. A surface-mounted control fence was positioned upstream of a backward-facing step, and as an oncoming flow a thin and fully developed turbulent boundary layer with a thickness of δ/h = 0.8 was used. The objective of the passive control was to enhance the entrainment rate of the shear layer bounding the separation zone behind the step, thereby reducing the mean reattachment length,〈 X r0 〉. Direct Numerical Simulations (DNS) and Large-Eddy Simulations (LES) at Re h = 3000 (based on the step height, h, and the free stream velocity, U ∞) were carried out for the uncontrolled and the controlled flow case. The LES results were in good agreement with the DNS reference solutions. Adaptively controlled feedback simulations showed that a certain minimum distance between the step edge and the upstream position of the control fence is required to achieve a maximum reduction of the reattachment length.

Exact solutions of problems of the three-dimensional flow of ideal viscous incompressible fluids near cylindrical surfaces

Three-dimensional flows of an incompressible fluid, the parameters of which depend on two coordinates and time, are considered. The stream surfaces of such flows are cylindrical. The equations of continuity and the Navier-Stokes equations can be transformed to relations, one of which is the equation for the stream function the other is the integral of the equations relating the pressure and the stream function, and the third is a linear equation for the projection of the velocity vector onto the axis parallel to the generatrix of the cylindrical surfaces. The problems of modelling the flows are considered on the basis of the exact solutions of the Navier-Stokes equations and Euler's equations using examples. Relations for the distribution of the flow parameters in the channel created by hyperbolical cylinders are derived for the case of unsteady inviscid flow. The streamlines of these flows are situated on the side surfaces of the hyperbolical cylinders and intercept the generatrices of the cylinders at certain indirect angles. The flow around a circular cylinder and the flow of fluid inside an elliptic cylinder are considered in the case of steady inviscid flow. The streamlines on the circular cylinder are arranged transverse to the cylinder (the projection of the velocity vector onto the coordinate axis, parallel to the generatrix of the cylinder, is equal to zero). Far from the cylinder the streamlines are also situated on a cylindrical surfaces, but not transverse to the cylinder, making certain indirect angles with the generatrix. Viscous three-dimensional flows, possessing a certain symmetry, are considered. In the case of radial symmetry the streamlines are helical lines. The non-planar Couette flow between parallel moving planes is characterized by the fact that the velocity vectors, being situated in the same plane, are collinear, while the velocity vectors in parallel planes are not collinear. Relations for viscous steady three-dimensional flows, using well-known relations, obtained for the stream function of two-dimensional flows, are given.

On the meaning and accuracy of the pressure–flow technique to determine constriction areas within the vocal tract

Since Warren and DuBois (D.W. Warren, A.B. DuBois, Cleft Palate Journal 1 (1964) 52–71), the “Pressure–Flow technique” has been widely used to estimate constriction areas within the vocal tract. In this paper, three fundamental questions regarding this technique are addressed: (1) What exactly is measured (minimum, maximum or “mean” areas)? (2) What degree of accuracy can be expected from this technique? (3) To what extent can this method be applied to unsteady flow conditions? A theoretical and experimental study based on a mechanical vocal tract model, including various constriction shapes, is presented. The pressure–flow technique is shown to be relatively insensitive to the exact constriction shape (circular, uniform or diverging), and the estimated area to be close to the minimum area of the constriction. This result can be theoretically rationalised by considering that in all cases studied here, the flow separation point is always close to the minimum constriction. Compared with much more complex viscous flow solutions, a simple one-dimensional flow model is shown to yield fair estimates of the areas (within 20%), except for low Reynolds numbers flows. The empirical head-loss factor, or flow coefficient, k=0.65, sometimes used, appears to be disputable and is probably due to an experimental artefact. Lastly these results are extended to the case of unsteady flow.

HEAT TRANSFER CHARACTERISTICS OF FLOW PAST A RECTANGULAR PROTRUDING BODY

Flow over a protruding body is an attractive research field in thermal engineering. In the present study, laminar flow over a protruding body is considered. Unsteady two-dimensional Navier-Strokes and energy equations are solved numerically using a control volume approach. The heat transfer characteristics due to vortex shedding are examined in detail. The flow and heat transfer characteristics are compared with their counterparts obtained from the steady flow case. The entropy analysis is carried out and irreversibility generated because of fluid friction is computed. The relative heat transfer and irreversibility ratios are introduced to compare the heat transfer performance characteristics of unsteady and steady flow cases. It is found that the relative heat transfer ratio attains higher value, whereas relative irreversibility ratio becomes less for unsteady flow as compared with that obtained for the steady flow case.

Three-dimensional dispersion of momentum in wave-induced nearshore currents

In this paper we consider the circulation induced by waves breaking near a coast. We show that the vertical nonuniformity of the wave-averaged horizontal velocities leads to mixing-like terms for the horizontal velocity in the depth-integrated equations of momentum. The mechanism is analogous to Taylor's (1953, 1954) shear-dispersion mechanism for solutes in a shear flow. The results presented here are an extension of the results found by Svendsen & Putrevu (1994) to the general case of unsteady flow over an arbitrary bottom topography.

Riemann Invariant Manifolds for the Multidimensional Euler Equations

A new approach for studying wave propagation phenomena in an inviscid gas is presented. This approach can be viewed as the extension of the method of characteristics to the general case of unsteady multidimensional flow. A family of spacetime manifolds is found on which an equivalent one-dimensional (1-D) problem holds. Their geometry depends on the spatial gradients of the flow, and they provide, locally, a convenient system of coordinate surfaces for spacetime. In the case of zero-entropy gradients, functions analogous to the Riemann invariants of 1-D gas dynamics can be introduced. These generalized Riemann invariants are constant on these manifolds and, thus, the manifolds are dubbed Riemann invariant manifolds (RIM). Explicit expressions for the local differential geometry of these manifolds can be found directly from the equations of motion. They can be space-like or time-like, depending on the flow gradients. This theory is used to develop a second-order unsplit monotonic upstream-centered scheme for conservation laws (MUSCL)-type scheme for the compressible Euler equations. The appropriate RIM are traced back in time, locally, in each cell. This procedure provides the states that are connected with equivalent 1-D problems. Furthermore, by assuming a linear variation of all quantities in each computational cell, it is possible to derive explicit formulas for the states used in the 1-D characteristic problem.

Inviscid-Viscous Coupled Solution for Unsteady Flows Through Vibrating Blades: Part 2—Computational Results

A quasi-three-dimensional inviscid-viscous coupled approached has been developed for unsteady flows around oscillating blades, as described in Part 1. To validate this method, calculations for several steady and unsteady flow cases with strong inviscid-viscous interactions are performed, and the results are compared with the corresponding experiments. Calculated results for unsteady flows around a biconvex cascade and a fan tip section highlight the necessity of including viscous effects in predictions of turbomachinery blade flutter at transonic flow conditions.

Stochastic analysis of dispersion in unsteady flow in heterogeneous aquifers

The dispersion of a solute plume resulting from unsteady flow in three‐dimensional, heterogeneous porous media was analyzed using stochastic continuum theory. Asymptotic stochastic solutions of the perturbed unsteady flow and solute transport equations were used to construct the macroscopic dispersive flux and evaluate the resulting macrodispersivity tensor in terms of a three‐dimensional, statistically anisotropic input covariance describing the hydraulic conductivity and an input covariance describing the temporal variability of the mean hydraulic gradient. The flow equation was approximated by neglecting the influence of internal storage due to medium and fluid compressibility (specific storage, SS ≈ 0). The predictive expression for the macrodispersivity tensor is the sum of two components: a spatial variability component identical to previous steady state theory and an unsteady component. Two special cases of unsteady flow were examined: variation only in the magnitude and variation only in the direction of the hydraulic gradient. Gradient magnitude variation produces a slightly larger longitudinal macrodispersivity than does the steady flow case, whereas gradient direction variation produces a significantly larger transverse macrodispersivity. Longitudinal and horizontal transverse macrodispersivities predicted with the unsteady stochastic theory were shown to be of a magnitude similar to observed values from the Borden, Cape Cod, and Columbus tracer experiments.

COMPUTATIONAL FLOW VISUALIZATION IN AN AORTIC ANEURYSM

Atherosclerosis and atherosclerotic aneurysms can occur in the abdominal aorta. Steady and pulsatile three-dimensional flow cases were simulated in an abdominal aortic aneurysm on a graphics workstation. In the steady flow case, in the aneurysm center, two symmetric vortices were formed, and flow separation started at the aneurysm inlet. Regions of high pressure were observed at the aneurysm exit caused by the symmetric jets that were formed, implying that this high-pressure region could be an area where rupture is most likely. The shear stress was low in the aneurysm portion of the vessel, and local maximum values were observed at the distal aneurysm constriction. In the unsteady flow case, the main vortex appeared and disappeared and changed position in the unsteady flow case and induced vortices were formed; in the steady flow case only one constant vortex was observed in the centerline view. The graphics system allows instant observation of the vascular structures from any angle or distance while moving, with wire frame and solid viewing modes being possible. The graphics programs allow the viewer to travel inside the aneurysm to observe the pressure and shear distribution in colorful detail, and videos have been made.

On the Prediction of Unsteady Forces on Gas Turbine Blades: Part 1—Description of the Approach

This article investigates the generation of unsteady forces on turbine blades due to potential-flow interaction and viscous-wake interaction from upstream blade rows. A computer program is used to calculate the unsteady forces on the rotor blades. Results are obtained by modeling the effects of the stator viscous wake and the stator potential-flow field on the rotor flow field. The results for one steady and one unsteady flow case are compared with known analytical and experimental data. The amplitudes for the two types of interaction are based on an analysis of available viscous wake data, on measurements of the potential-flow disturbance downstream of typical turbine stators, and on a parametric study of the effects of the amplitudes on the results of the unsteady forces generated on a typical turbine rotor cascade.

The two-dimensional vortex motions of a viscous incompressible fluid

A generalization of the Helmholtz theorems concerning the conservation of vortex lines and the intensity of vortex tubes, which are well known in the hydrodynamics of a non-viscous fluid, and of Kelvin's theorem on the conservation of the circulation of the velocity along a closed fluid contour is presented for the case of unsteady planar and axisymmetrical continuous flows of a viscous incompressible fluid.

Flow instability of molten silicon in the Czochralski configuration

The flow instability of molten silicon in the Czochralski configuration has been studied by in-situ observation of melt convection using X-ray radiography and by temperature fluctuation measurement during crystal growth. Flow mode was dependent on an aspect ratio of the melt. For a deep, low aspect ratio melt, with growing crystal which is identical to shouldering process of the growth, the flow was unsteady and non-axisymmetric. For a shallow melt without crystal and crucible rotations, the flow was relatively steady and axisymmetric. However, flow became unsteady and non-axisymmetric for a shallow melt with crystal rotation. Amplitude of directly measured temperature fluctuation in the molten silicon for the case of unsteady and non-axisymmetric flow was larger than that for the relatively steady and axisymmetric flow. The flow instability area, which was also thermally unstable, was found to be larger in the crystal/crucible iso-rotation condition. In contrast Munakata and Tanasawa reported at the International Symposium on Supercomputers for Mechanical Engineering, September 1988, Tokyo that flow instability area was small for the silicone oil with larger Prandtl number.

Conjugated Heat Transfer From a Hot-Film Probe for Transient Air Flow

A numerical heat transfer model of a flush-mounted hot-film probe is presented for transient air flow. The model geometry involves simultaneous heat flow in both the air and the adjacent slab wall. Two cases are considered, steady laminar flow and laminar flow with a low-frequency oscillating component. In the steady-flow case, the numerical results compare favorably with steady calibration data from the literature. In the unsteady flow case, transfer function results are consistent with literature experimental data. In both cases, agreement with the literature data is obtained by calibrating the numerical model at zero flow.

Some cases of unsteady flow in porous media associated with irrigation

The results of an earlier study [1] are used to consider the motion of underground water flowing from an irrigation channel. The surface of the ground flow usually has an uneven shape, since in some sections there is stripwise irrigation, while at others evaporation may occur or rain may fall. The variations of the water level in the channel are assumed known. The subsequent variation of the ground water level is determined. If the level rises appreciably there can be salinization or swamped ground can form.

Characteristics of pulsation error on constriction flowmeterings in compressible flow.

In order to clarify the measuring error on differential-pressure-type flowmetering under the conditions of compressible flow with pulsation, the effects of fluid compressibility, beta ratio, upstream pipe length, pulsation intensity and Strouhal number on the error have been investigated respectively, by experiment and unsteady compressible flow analysis. The characteristics of the error E with the Strouhal number, St, are plotted indirectly and compactly, by introducing the variation of MR (ratio of mass flow amplitude in an unsteady flow case to that of quasi steady flow case) with St. This curve makes it possible to estimate the error curves qualitatively.

Exact solutions of the unsteady two-dimensional Navier-Stokes equations

This paper studies the two-dimensional incompressible viscous flow in which the local vorticity is proportional to the stream function perturbed by a uniform stream. It was known by Taylor and Kovasznay that the Navier-Stokes equations for flow of this kind become linear. From the general solution to the linear equations for steady flow, we show that there exist only two types of steady flow of this kind: Kovasznay downstream flow of a two-dimensional grid and Lin and Tobak reversed flow about a flat plate with suction. In the unsteady flow case, new classes of exact analytical solutions are found which include Taylor vortex array solution as a special case. It is shown that these unsteady flows are, as viewed from a frame of reference moving with the undisturbed uniform stream, pseudo-steady in the sense that the flow pattern is steady but the magnitude of motion decays, or grows, exponentially in time. All these solutions are valid for any Reynolds number.

Mathematical simulation of unsteady flow past a wing with jets

The various approximate approaches to the investigation of the unsteady aerodynamic characteristics of an airfoil with jet flap [1–3] are applicable only for an airfoil, low jet intensity, and low oscillation frequencies. In the present paper, the method of discrete vortices [4] is generalized to the case of unsteady flow past a wing with jets and arbitrary shape in plan. The problem is solved in the linear formulation; the conditions used are standard: no flow through the wing and jet, finite velocities at the trailing edges where there is no jet, and also a dynamical condition on the jet. The wing and jet are assumed to be thin and the medium inviscid and incompressible.

Unsteady supersonic flow around blunt bodies with detached shock wave

The numerical method of calculating the supersonic three-dimensional flow about blunt bodies with detached shock wave presented in [1–3] is applied to the case of unsteady flow. The formulation of the unsteady problem is analogous to that of [4], which assumes smallness of the unsteady disturbances. The paper presents some results of a study of the unsteady flow about blunt bodies over a wide range of variation of the Mach number M∞=1.50−∞ and dimensionless oscillation frequency ωl/V∞=0−1.0. A comparison is made with the results obtained from the Newton theory.