Showing papers on "Stream function published in 1988"

The Generation of Global Rotational Flow by Steady Idealized Tropical Divergence

Abstract: Tropical convective heating is balanced on the large scale by the adiabatic cooling of ascent. The horizontal divergence of the wind above this heating may be viewed as driving the upper tropospheric rotational wind field. A vorticity equation model is used to diagnose this relationship. It is shown that because of the advection of vorticity by the divergent component of the flow, the Rossby wave source can be very different from the simple −fD source often used. In particular, an equatorial region of divergence situated in easterly winds can lead to a Rossby wave source in the subtropical westerlies where it is extremely effective. This part of the source can be relatively insensitive to the longitudinal position of the equatorial divergence. A divergence field which is asymmetric about the equator can lead to a quite symmetric Rossby wave source. For a steady frictionless flow the Rossby wave source averaged over regions within closed streamfunction or absolute vorticity contours is, under cert...

Integral formulation for 3D eddy-current computation using edge elements

Abstract: An integral formulation for eddy-current problems in nomagnetic structures is presented. The solenoidality of the current density is assured by introducing a current vector potential T. This potential possesses only two scalar components, as the gauge chosen to ensure its uniqueness is T. u = 0, where u is a prescribed vector field. The discrete analogue of this gauge and the boundary conditions are directly imposed by the shape functions, exploiting the use of edge finite elements and the methods of network theory. In the frame of the integral methods, this approach seems the most adequate to analyse the eddy currents induced in both massive conductors and thin shells. In massive structures, the two degrees of freedom are to be compared to four of the usual integral methods which exploit the presence of a scalar potential to ensure solenoidality. On the other hand, the procedure naturally reduces to the stream function approach when applied to thin shells. Finally, an integration procedure which guarantees symmetry and positive-definiteness of the inductance matrix is proposed.

Instability and multiple steady states in a meridional-plane model of the thermohaline circulation

Abstract: A meridional-plane model of the thermohaline circulation with a simple friction force and advection and vertical diffusion of the T-S field has been used to demonstrate the instability and existence of multiple steady states associated with “mixed” T-S boundary conditions (specified temperature, flux condition for salinity). With forcing and geometry symmetric to the equator, the symmetric solution was found to be unstable to infinitesimal perturbations, and an asymmetric pole-to-pole circulation was the end-result in all cases. The structure obtained for the meridional-plane stream function and for the poleward heat flux are in qualitative agreement with those obtained by Bryan (1986). Convective overturning caused by static instability was not found to be essential for the transition to the asymmetric steady state. The study suggests that certain aspects of the ocean circulation, in particular those related to the ocean climate, may be profitably explored by use of two-dimensional, zonally averaged models. DOI: 10.1111/j.1600-0870.1988.tb00414.x

Experimental characterization of steady two-dimensional vortex couples

Abstract: We study the evolution of unsteady two-dimensional vorticity structures surrounded by fluid at rest. The flow is initiated by a short fluid impulse in a horizontal layer of mercury and is constrained to be two-dimensional by a vertical uniform magnetic field. The impulse is generated by an electric pulse between two electrodes, and a flow circulation can be produced by diverting part of the current through the external frame. The velocity field is measured from the streaks of small particles floating on the free upper surface, and the vorticity is then obtained by means of an analytical interpolation and differentiation. The flow always evolves toward a set of independent steady structures with symmetry which are either circular vortices (monopoles) or couples (dipoles). The latter have a linear or circular steady motion depending on the flow circulation around them. The region of non-zero vorticity is always close to a circle. The steadiness is confirmed by plotting the vorticity versus the stream function in the frame of reference moving with the couple. We obtain a curve, as appropriate for a steady solution of the Euler equation. The slope of this curve is either constant or has no maximum. We suggest that this result could correspond to a general stability condition. The interaction between two symmetric couples at various angles of incidence yields two new couples by exchange of their vortices. Oscillations of the resulting couples are often damped by releasing a circular vortex.

Flow in a channel with a moving indentation

Abstract: The unsteady flow of a viscous, incompressible fluid in a channel with a moving indentation in one wall has been studied by numerical solution of the Navier-Stokes equations. The solution was obtained in stream-function-vorticity form using finite differences. Leapfrog time-differencing and the Dufort-Frankel substitution were used in the vorticity transport equation, and the Poisson equation for the stream function was solved by multigrid methods. In order to resolve the boundary-condition difficulties arising from the presence of the moving wall, a time-dependent transformation was applied, complicating the equations but ensuring that the computational domain remained a fixed rectangle.Downstream of the moving indentation, the flow in the centre of the channel becomes wavy, and eddies are formed between the wave crests/troughs and the walls. Subsequently, certain of these eddies ‘double’, that is a second vortex centre appears upstream of the first. These observations are qualitatively similar to previous experimental findings (Stephanoff et al. 1983, and Pedley & Stephanoff 1985), and quantitative comparisons are also shown to be favourable. Plots of vorticity contours confirm that the wave generation process is essentially inviscid and reveal the vorticity dynamics of eddy doubling, in which viscous diffusion and advection are important at different stages. The maximum magnitude of wall vorticity is found to be much larger than in quasi-steady flow, with possibly important biomedical implications.

Resin Impregnation During the Manufacturing of Composite Materials Subject to Prescribed Injection Rate

Abstract: A numerical investigation into the area of resin impregnation during the manufacturing of composite materials is undertaken using a macroscopic flow through porous media ap proach. This study is specifically directed at modeling the resin transfer molding and resin film stacking advanced composite manufacturing processes. Quasi-steady state isothermal flow is assumed and a two-dimensional Darcy Law based stream function formulation is utilized. The resultant single governing equation for each quasi-steady timestep is solved along with proper boundary conditions using the method of boundary-fitted coordinate systems encompassing numerical grid generation. The resultant code is validated by a comparison with previously published results for flow into a rectangular mold with a point source and a line sink. Streamlines, pressure distributions, velocity profiles, and temporal liquid free surface positions are then presented for the flow into a mold of general irregular geometry encasing both isotropic and anis...

Steady symmetric vortex pairs and rearrangements

Abstract: We prove an existence theorem for a steady planar flow of an ideal fluid, containing a bounded symmetric pair of vortices, and approaching a uniform flow at infinity. The data prescribed are the rearrangement class of the vorticity field, and either the momentum impulse of the vortex pair, or the velocity of the vortex pair relative to the fluid at infinity. The stream function ψ for the flow satisfies the semilinear elliptic equationin a half-plane bounded by the line of symmetry, where φ is an increasing function that is unknown a priori. The results are proved by maximising the kinetic energy over all flows whose vorticity fields are rearrangements of a specified function.

Petrov‐Galerkin methods on multiply connected domains for the vorticity‐stream function formulation of the incompressible Navier‐Stokes equations

Abstract: SUMMARY In this paper we present streamline-upwind/Petrov-Galerkin finite element procedures for two-dimensional fluid dynamics computations based on the vorticity-stream function formulation of the incompressible Navier-Stokes equations. We address the difficulties associated with the convection term in the vorticity transport equation, lack of boundary condition for the vorticity at no-slip boundaries, and determination of the value of the stream function at the internal boundaries for multiply connected domains. The proposed techniques, implemented within the framework of block-iteration methods, have successfully been applied to various problems involving simply and multiply connected domains. There are some advantages in using the vorticity-stream function formulation of the incompressible Navier-Stokes equations for two-dimensional computations. Compared to the velocity-pressure formulation, the vorticity-stream function formulation leads to computed flow fields which satisfy the incompressibility condition automatically; also the number of unknown functions is reduced from three to two and the vorticity field is computed directly instead of being obtained by differentiation of the velocity field. The last advantage becomes important if one needs to study the vorticity field and therefore wants that this field be represented as accurately as possible. We propose suitable finite element procedures for the solution of the time-dependent vorticity transport equation and the Poisson’s equation which relates the stream function to the vorticity. The difficulties associated with the convection term in the vorticity transport equation, lack of

Numerical simulation of the buoyancy-induced flow in a partially open enclosure

Abstract: A numerical study of a buoyancy-induced flow generated by a finite-size heat source located on a vertical wall of an enclosure with a single opening is carried out. A two-dimensional laminar flow is assumed. Employing a stream function vorticity formulation within the framework of the Boussinesq approximations, the temperature and the flow fields are computed. The effect of the opening on the induced flow is investigated in terms of the size and the location of the opening. Numerical results are obtained for a wide range of governing parameters, particularly the Rayleigh number. The Prandtl number is taken as that corresponding to air at normal con ditions, Pr = 0.72, and the aspect ratio of the enclosure is varied. Of particular interest were the flow generated in the vicinity of the opening, the flow adjacent to the heated surface, and any stratification that might arise in the enclosure. All these and several other relevant aspects are considered in this study. The numerical formulation for the boundar...

Deducing the Wind from Vorticity and Divergence

Abstract: The horizontal wind field may be deduced from the vorticity and divergence by solving Poisson equations for the velocity potential and streamfunction or, more directly, by the solution of a single Poisson equally accurate. If the domain is of limited extent, boundary conditions must be specified. It is sufficient to prescribe a single component of the boundary velocity. Methods which use both components overdetermine the solution and may not converge in general.

The decay of a viscous vortex pair

Abstract: The evolution of a viscous vortex pair is investigated through the use of a heuristic model. The model is based on the linear superposition of two Oseen vortices of opposite circulation spaced a distance 2b apart. The vortices are allowed to evolve through viscous diffusion and their mutual induction. The motion is unforced and as a consequence the total hydrodynamic impulse is exactly conserved for all time. In the model the total circulation in the upper half plane is assumed to remain initially constant. This constraint is applied up to a finite time when the model solution reaches its asymptotic form corresponding to a drifting Stokes dipole dominated by interdiffusion of vorticity across the plane of symmetry. The drift velocity of the vortex pair is determined by the condition that the integrated pressure force vanishes on the line of symmetry at all times. At large time this leads to an asymptotic value of the drift velocity which scales with the similarity properties of the Stokes solution. To provide a more rigorous foundation for the drift, the asymptotic behavior of the flow for large time is investigated through an expansion of the solution in inverse powers of the time. First the second‐order pressure is determined as a solution of a Poisson equation with the source term generated by the first‐order flow field. Surprisingly, the solution turns out to be independent of the drift. Nevertheless, an exact condition for the drift is found by considering the limiting form of the second‐order pressure at infinity where the flow is irrotational and the pressure can be computed directly from the first‐order velocity field using Bernoulli’s equation. In this latter approach the far field pressure is determined up to an unknown function of time which upon comparison with the Poisson solution is identified as the drift. The exact drift obtained in this fashion differs by only 10% from the value obtained using the pressure field of the heuristic model. Finally, it is shown that the existence of the complete second‐order asymptotic solution of the Navier–Stokes equations requires the inclusion of the same drift in the first‐order solution that was found from the examination of the pressure. The second‐order vorticity and streamfunction are determined; the latter contains afree constant to accommodate conditions at earlier times. Prospects for the existence of higher‐order asymptotic terms are discussed.

Geostrophic Vortex Dynamics

Abstract: By generalizing the method of contour dynamics to the quasigeostrophic two layer model, we have proposed and solved a number of fundamental problems in the dynamics of rotating and stratified vorticity fields. A variety of rotating and translating potential vorticity equilibria (V-states) in one and two layers have been obtained, shedding new light on potential vorticity dynamics in the geostrophic context. In particular,the equivalent barotropic model is shown to be a singular limit of the two-layer model for scales large compared to the radius of deformation. The question of coalescence of two vortices in the same layer (merger) and· in different layers (alignment) is studied in detail. Critical initial separation distances for coalescence are numerically established as functions of the radius of deformation and the relative thickness of the layers at rest. The connection between coalescence and the existence of stable rotating doubly-connected V-states is shown to be an illuminating generalization of the Euler results. The question of filamentation of two-dimensional vorticity interfaces is addressed from a new geometrical perspective. The analysis of the topology of the streamfunction in a frame of reference rotating with the instantaneous angular velocity of the vorticity distribution (the corotating frame) is shown to yield new powerful insights on the nonlinear evolution of the vorticity field. In particular, the presence of hyperbolic (critical) points of the corotating streamfunction that come in contact with the vorticity interface is found to be directly responsible for the generation of filaments. The importance ofthe position of the critical points of the comoving streamfunction is found to generalize to the two-layer quasigeostrophic context. They are shown to play the crucial role in determining the limits, in parameter space, on the existence of a number of two-layer rotating and translating potential vorticity equilibria.

Entropy and vorticity corrections for transonic flows

Abstract: Different models for inviscid transonic flows are examined. The common assumptions that the flow is isentropic and irrotational are critically evaluated. Entropy and vorticity correction procedures for potential and stream function formulations are presented, together with the details of the treatment of shocks and wakes, and drag and lift calculations. The non-uniqueness problem of the potential formulation is studied using different artificial viscosity forms. Numerical results are compared with Euler solutions.

A global branch of steady vortex rings.

Abstract: : A steady vortex ring of prescribed strength and propagation speed can be described in terms of a Strokes stream function psi. A flux constant k measures the flow through the center of the axisymmetric vortex ring. For k = 0, Hill in 1984 found an explicit solution for the semi-linear elliptic equation satisfied by psi. In this paper it is shown that there is an unbounded, closed, connected branch of solutions emanating from Hill's vortex in the space of pairs (k, psi). (Author)

On finite element approximations of the streamfunction-vorticity and velocity-vorticity equations

Abstract: We consider finite element methods for vorticity formulations of viscous incompressible flows. In two-dimensional settings the familiar streamfunction-vorticity formulation is examined. We focus on its accuracy, especially when using low-order elements, and on its use with a variety of boundary conditions and in multiply connected domains. In three dimensions the velocity-vorticity formulation is shown to be preferable, and a promising algorithm using this formulation is presented. We close by considering the recovery of the pressure field once the streamfunction or velocity fields are known. In particular we describe and analyse an algorithm for recovering the pressure which is based on well known methods for the primitive variable formulation and which requires no boundary conditions on the pressure at solid walls.

Parallel flow convection in a tilted two‐dimensional porous layer heated from all sides

Abstract: In this work natural convection within a two‐dimensional fluid saturated Darcy porous layer is considered. The porous material is in a large aspect ratio rectangle with its sides inclined with respect to the gravity vector. All four faces are exposed to uniform heat fluxes, opposite faces being heated and cooled, respectively. Analytical solutions for the streamfunction and temperature fields in the central region are deduced using a parallel flow assumption. Numerical confirmation of the analytical results is also obtained. For high enough Rayleigh numbers multiple steady states around the rest state are found. These states represent unsymmetrical flows in opposite directions. The critical Rayleigh number for the onset of motion determined from a stability analysis corresponds to that for the existence of unicellular convection using the parallel flow approximation. Linear stability limits of each of the multiple states are also calculated. One of the convective flows is found to be stable to higher Rayleigh numbers than the other.

Computation of three dimensional transonic flows using two stream functions

Abstract: A computational method for three-dimensional flows is presented in terms of two stream functions, which may be considered as two components of a generalized vector potential. An iterative scheme is developed such that only a sequence of two-dimensional-like problems, for each function, is solved. The convergence of the iterative scheme is studied based on von Neumann linear analysis. For transonic flow calculation, numerical methods used for potential flows are readily applied, namely artificial density and Zebra relaxation. Results of transonic flow calculations around a wing are presented.

Stagnation Point Flow of a Non-Newtonian Second Order Fluid

Abstract: In this paper the flow near a two-dimensional stagnation point for a particular non-Newtonian fluid has been studied. For a second order fluid the equation of motion for the stream function has bee...

On the similarity of velocity and temperature profiles in strong (variable density) turbulent buoyant plumes

Abstract: Various similarity hypothesis for axisymmetric turbulent buoyant plumes with large density differences (strong plumes) are discussed in this paper, In contrast to buoyant flows having small density variations (Boussinesqu limit), controversies remain in modeling strong buoyant flows for which (▵e/e0=(e0-e)/e0=O(1)), We review: 1) previous attempts to model such flows. 2) experimental evidence that the flow properties for strong plumes have similar profiles. and 3) a consistent derivation of the similarity relations. A significant conclusion from this analysis is that the similarity profiles for velocity and density defect (or temperature rise) should be expressed in terms of an average stream function or equivalently the Howarth variable (e.g.,n=∫0 r(e/e0rdr for axisymmetric plumes) which includes variable density effects. Others have suggested, instead, that the plume property profiles might be expressed by using a laboratory radial coordinate normalized by a local characteristic flow width, in ...

Reentrant corner singularities in non-newtonian flow. Part I: theory

Abstract: Biorthogonal series expansions are used to study the creeping flow of a corotational Maxwell fluid in plane reentrant and non-reentrant sectors. The governing equations are formulated in terms of stream function and Airy stress function only, the radial parts of which are determined by an infinite system of fourth-order nonlinear ordinary differential equations which are seen to be singular perturbations of the linear second-order equations for Stokes flow. Exact formal solutions in logarithmic series are derived, but do not readily enable the asymptotics of the corner singularities to be studied. Instead, approximate methods based on Picard linearization and dominant component analysis are used to decouple the system, whence the singularities are studied in terms of generalized hypergeometric equations and Meijer G-functions. Certain conditions on the roots of associated quartic equations are derived, the satisfaction of which would indicate the existence of lip vortices in a reentrant sector and the integrability of stress in a neighbourhood of the corner.

Diffusion of a ring vortex

Abstract: The diffusion of a ring vortex is investigated in the present paper with allowance for the influence of the initial radius of the toroidal vorticity distribution on the flow structure. The statement of the problem in such a formulation makes it possible to classify and reinterpret results obtained previously. A vortex pair is studied together with a vortex ring. The toroidal vorticity and stream function distributions are obtained analytically. The self-induced lift velocity of the ring vortex is found. The influence of inertial terms is investigated numerically.

Topographic Rossby waves above a random array of seamountains

Abstract: The barotropic potential vorticity equation or topographic wave equation is not linear in topography: the solution structure for topography formed from a sum of obstacles is not the sum of solutions for the obstacles in isolation, even when these individual solutions have identical frequencies. This paper considers the mechanism by which normal modes of oscillation above one mountain are modified by interactions with its neighbours. Exact explicit solutions for the normal modes above a pair of circular seamountains show that the interactions between the mountains rapidly approaches the large-separation approximation obtained by considering solely the first reflection of the disturbance of one mountain at the other. For mountains of one diameter separation at the closest point, the approximation is accurate to within 1%. Perhaps surprisingly, coupling between two identical mountains is weak and resonance occurs between mountains and dales of equal and opposite height. The accurate approximate solutions enable consideration of the effects on a mountain of an infinite set of randomly distributed neighbours. The ensembleaveraged frequency for a mountain of given height is determined in terms of the area fraction of the other mountains. The idea of an effective topography is introduced for the ensemble-averaged stream function: it is that (non-random) topography generating a stream function identical to the ensemble-averaged stream function. This differs markedly from the ensemble-averaged topography. The explicit form of the effective topography is derived for a set of right circular cylinders.

Kinematics and return flow in a closed wave flume

Abstract: Stokes (1847) showed that finite amplitude progressing waves cause a net drift of fluid, in the direction of wave motion, which occurs in the upper portion of the water column. In a closed wave flume this drift must be accompanied by a return flow toward the wave generator to satisfy the conservation of mass. This study presents Eulerian velocity and water surface measurements soon after the onset of wave motion from 12 locations in a large scale flume. Waves with .67 < kh < 2.29 and .09 < H/h < .39 were produced in a water depth of 3.5 meters. Superimposing the return flow theory of Kim (1984) with seventh order stream function theory is shown to improve the velocity predictions. The measured return flows are a function of time and depth and agree with Kim's theory as a first approximation. The mean water surface set-down agrees with the theory of Brevik (1979) except for the nearly deep water waves.

A skewed eddy of Batchelor-modon type

Abstract: A Batchelor-modon eddy is a highly specialized nonlinear vortex pair, whose potential vorticity depends linearly on the stream function viewed from the coordinates moving with the translation velocity of the eddy. To generalize it, a skewed model is developed by introducing a cubic nonlinearity in addition to the linear term.

Numerical analysis of multiple element high lift devices by Navier Stokes equation using implicit TVD finite volume method

Abstract: This paper deals with the analysis of multiple element high lift devices by solving the Navier Stokes equations using the TVD(Tota1 Variation Diminishing) finite difference method. In order to generate a computational grid around the multiple element airfoils automatically, the grid generator using the elliptic method, in which Poisson equations are by the finite difference method, combined with 2-D panel method is developed. As to the flow solver, some improvements are added to the TVD scheme to calculate low Mach number flows efficiently. Numerical calculations are carried out for the single slotted flap configuration. Multiple element H.L.D.(High Lift Devices) are commonly used in many transport aircrafts to obtain high lift forces necessary in landing and taking off. In designing H.L.D., however, the accurate estimation of their performances by wind tunnel testings is a difficult problem since high Reynolds number flows are hard to obtain in a wind tunnel. The analysis by CFD(Computationa1 Fluid Dynamics) is thus expected as a powerful tool in designing H.L.D. Recent advancements of super computers and numerical algorithms have made it possible to use the finite difference calculation of Navier-Stokes equations as a practical design tool, but some problems are still remained to use it for predicting the performance of H.L.D. Among these problems, the most urgent problem is to establish an automated numerical grid generation method around the multiple element airfoils. In the present work, we develop an automated numerical grid generator for multiple airfoils utilizing the elliptic method combined with 2 dimensional panel method and an implicit TVD(Tota1 Variation Diminishing) scheme suited for analyzing the flow field of multiple element H.L.D. The finite difference calculation of Navier Stokes equations for H.L.D. using these methods are carried out. Calculated results compared with experimental data are shown in the latter sections. Research engineer. 2.GRID GENERATION In order to calculate the flow around multiple element high lift devices by a finite difference method, it is necessary to generate a computational grid around multiple body configuration. As is usual to move each element in parametric design process of high lift devices , the numerical grid for each configuration is also needed. Thus, in order to use the Navier-Stokes analysis as a practical design tool, it is highly desirable that the grid generator can treat each configuration automatically. We utilize the solution of potential flow around airfoils as a building block of the automated grid generator. Since the dividing stream lines contain body surface and the potential lines are orthogonal to those stream lines, we can make orthogonal body fitted coordinate around airfoils. Using a panel method, a solution of the potential flow around arbitrary multiple bodies can be obtained so that we can generate a grid system aut~rnaticall~.[ ']~[~] The stream function Ij, and the potential qhof the p e tential flow are governed by following differential equations and boundary conditions: nldx + nz& = 0 on the body surfaces, (2.3) 4 = const on the body surfaces, (2.4) where (n l , n z ) is a unit vector normal to the body surfaces. The solution of these equations is easily obtained by the panel method. Now consider the strip region enclosed by two dividing stream lines and two potential lines. The stream function and the potential in this region are governed by equations(2.1) to (2.4) and the additinal boundary conditions: n l& + n2du = 0 on the strean lines, (2.5) $J = const on the stream lines, (2.6) nl$, + n2& = 0 on the potential lines, (2.7) $ = const on the potential lines, (2.8) where (nl, nz) is a unit vector normal to a stream line or a potential line. Although it is possible to generate a grid by tracing stream lines and potential lines in principle, the algorithm becomes complicated and the accuracy of panel method is not sufficient near the bodv surface to generate fine grids appropriate for Navier Stokes calculation. Therefore the following indirect method is adopted instead. After Thompson, Thames and Mastin 13],equations (2.1) and (2.3) can be transformed as follows:

Mixed Laminar Convection in Trombe Wall Channels

Abstract: The two-dimensional, steady, combined forced and natural convection in a vertical channel is investigated for the laminar regime. To simulate the Trombe wall channel geometry properly, horizontal inlet and exit segments have been added to the vertical channel. The vertical walls of the channel are maintained at constant but different temperatures while the horizontal walls are insulated. A finite difference method using up-wind differencing for the nonlinear convective terms, and central differencing for the second order derivatives, is employed to solve the governing differential equations for the mass, momentum, and energy balances. The solution is obtained for stream function, vorticity, and temperature as the dependent variables by an iterative technique known as successive substitution with overrelaxation. The flow and temperature patterns in the channel are obtained for Reynolds numbers and Grashof numbers ranging from 25 to 100 and 10,000 to 1,000,000, respectively. Both local and overall heat transfer coefficients are computed for the channel aspect ratio varying from 5 to 15. For a given value of Grashof number, as the Reynolds number is increased, the flow patterns in the vertical channel exhibit a change from natural convection like flow patterns in which a large recirculating region is formed in themore » vertical part of the channel, to a forced flow type pattern. This is also the case with isotherms. The size of the recirculating region in the channel increases with increasing value of Gr/Re/sup 2/. At low Reynolds number, the stream function, and isotherms are qualitatively similar to those reported for the natural convection in rectangular slots.« less

Bed load transport due to non-linear wave motion

Abstract: The bed load transport rate due to wave motion is measured in a wave flume. The modified stream function theory of the author ( Tanaka (1988) ) is applied to the formulation of the sediment transport rate in order to include the non-linearity. The proposed formula predicts well except near the surf zone where the effect of the acceleration plays an important role.