Showing papers on "Stream function published in 1986"

Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms

Abstract: I. Mathematical Foundation of the Stokes Problem.- 1. Generalities on Some Elliptic Boundary Value Problems.- 1.1. Basic Concepts on Sobolev Spaces.- 1.2. Abstract Elliptic Theory.- 1.3. Example 1: Dirichlet's Problem for the Laplace Operator.- 1.4. Example 2: Neumann's Problem for the Laplace Operator.- 1.5. Example 3: Dirichlet's Problem for the Biharmonic Operator.- 2. Function Spaces for the Stokes Problem.- 2.1. Preliminary Results.- 2.2. Some Properties of Spaces Related to the Divergence Operator.- 2.3. Some Properties of Spaces Related to the Curl Operator.- 3. A Decomposition of Vector Fields.- 3.1. Decomposition of Two-Dimensional Vector Fields.- 3.2. Application to the Regularity of Functions of H(div ?) ? H(curl ?).- 3.3. Decomposition of Three-Dimensional Vector Fields.- 3.4. The Imbedding of H(div ?) ? H0 (curl ?) into H1(?)3.- 3.5. The Imbedding of H0(div ?) ? H (curl ?) into H1(?)3.- 4. Analysis of an Abstract Variational Problem.- 4.1. A General Result.- 4.2. A Saddle-Point Approach.- 4.3. Approximation by Regularization or Penalty.- 4.4. Iterative Methods of Gradient Type.- 5. The Stokes Equations.- 5.1. The Dirichlet Problem in the Velocity-Pressure Formulation.- 5.2. The Stream Function Formulation of the Dirichlet Problem in Two Dimensions.- 5.3. The Three-Dimensional Case.- Appendix A. Results of Standard Finite Element Approximation.- A.l. Triangular Finite Elements.- A.2. Quadrilateral Finite Elements.- A.3. Interpolation of Discontinuous Functions.- II. Numerical Solution of the Stokes Problem in the Primitive Variables.- 1. General Approximation.- 1.1. An Abstract Approximation Result.- 1.2. Decoupling the Computation of uh and ?h.- 1.3. Application to the Homogeneous Stokes Problem.- 1.4. Checking the inf-sup Condition.- 2. Simplicial Finite Element Methods Using Discontinuous Pressures.- 2.1. A First Order Approximation on Triangular Elements.- 2.2. Higher-Order Approximation on Triangular Elements.- 2.3. The Three-Dimensional case: First and Higher-Order Schemes.- 3. Quadrilateral Finite Element Methods Using Discontinuous Pressures.- 3.1. A quadrilateral Finite Element of Order One.- 3.2. Higher-Order Quadrilateral Elements.- 3.3. An Example of Checkerboard Instability: the Q1 - P0 Element.- 3.4. Error Estimates for the Q1 - P0 Element.- 4. Continuous Approximation of the Pressure.- 4.1. A First Order Method: the "Mini" Finite Element.- 4.2. The "Hood-Taylor" Finite Element Method.- 4.3. The "Glowinski-Pironneau" Finite Element Method.- 4.4. Implementation of the Glowinski-Pironneau Scheme.- III. Incompressible Mixed Finite Element Methods for Solving the Stokes Problem.- 1. Mixed Approximation of an Abstract Problem.- 1.1. A Mixed Variational Problem.- 1.2. Abstract Mixed Approximation.- 2. The "Stream Function-Vorticity-Pressure" Method for the Stokes Problem in Two Dimensions.- 2.1. A Mixed Formulation.- 2.2. Mixed Approximation and Application to Finite Elements of Degree l.- 2.3. The Technique of Mesh-Dependent Norms.- 3. Further Topics on the "Stream Function-Vorticity-Pressure" Scheme.- 3.1. Refinement of the Error Analysis.- 3.2. Super Convergence Using Quadrilateral Finite Elements of Degree l.- 4. A "Stream Function-Gradient of Velocity Tensor" Method in Two Dimensions.- 4.1. The Hellan-Herrmann-Johnson Formulation.- 4.2. Approximation with Triangular Finite Elements of Degree l.- 4.3. Additional Results for the Hellan-Herrmann-Johnson Scheme.- 4.4. Discontinuous Approximation of the Pressure.- 5. A "Vector Potential-Vorticity" Scheme in Three Dimensions.- 5.1. A Mixed Formulation of the Three-Dimensional Stokes Problem.- 5.2. Mixed Approximation in H(curl ?).- 5.3. A Family of Conforming Finite Elements in H(curl ?).- 5.4. Error Analysis for Finite Elements of Degree l.- 5.5. Discontinuous Approximation of the Pressure.- IV. Theory and Approximation of the Navier-Stokes Problem.- 1. A Class of Nonlinear Problems.- s Problem for the Laplace Operator.- 1.5. Example 3: Dirichlet's Problem for the Biharmonic Operator.- 2. Function Spaces for the Stokes Problem.- 2.1. Preliminary Results.- 2.2. Some Properties of Spaces Related to the Divergence Operator.- 2.3. Some Properties of Spaces Related to the Curl Operator.- 3. A Decomposition of Vector Fields.- 3.1. Decomposition of Two-Dimensional Vector Fields.- 3.2. Application to the Regularity of Functions of H(div ?) ? H(curl ?).- 3.3. Decomposition of Three-Dimensional Vector Fields.- 3.4. The Imbedding of H(div ?) ? H0 (curl ?) into H1(?)3.- 3.5. The Imbedding of H0(div ?) ? H (curl ?) into H1(?)3.- 4. Analysis of an Abstract Variational Problem.- 4.1. A General Result.- 4.2. A Saddle-Point Approach.- 4.3. Approximation by Regularization or Penalty.- 4.4. Iterative Methods of Gradient Type.- 5. The Stokes Equations.- 5.1. The Dirichlet Problem in the Velocity-Pressure Formulation.- 5.2. The Stream Function Formulation of the Dirichlet Problem in Two Dimensions.- 5.3. The Three-Dimensional Case.- Appendix A. Results of Standard Finite Element Approximation.- A.l. Triangular Finite Elements.- A.2. Quadrilateral Finite Elements.- A.3. Interpolation of Discontinuous Functions.- II. Numerical Solution of the Stokes Problem in the Primitive Variables.- 1. General Approximation.- 1.1. An Abstract Approximation Result.- 1.2. Decoupling the Computation of uh and ?h.- 1.3. Application to the Homogeneous Stokes Problem.- 1.4. Checking the inf-sup Condition.- 2. Simplicial Finite Element Methods Using Discontinuous Pressures.- 2.1. A First Order Approximation on Triangular Elements.- 2.2. Higher-Order Approximation on Triangular Elements.- 2.3. The Three-Dimensional case: First and Higher-Order Schemes.- 3. Quadrilateral Finite Element Methods Using Discontinuous Pressures.- 3.1. A quadrilateral Finite Element of Order One.- 3.2. Higher-Order Quadrilateral Elements.- 3.3. An Example of Checkerboard Instability: the Q1 - P0 Element.- 3.4. Error Estimates for the Q1 - P0 Element.- 4. Continuous Approximation of the Pressure.- 4.1. A First Order Method: the "Mini" Finite Element.- 4.2. The "Hood-Taylor" Finite Element Method.- 4.3. The "Glowinski-Pironneau" Finite Element Method.- 4.4. Implementation of the Glowinski-Pironneau Scheme.- III. Incompressible Mixed Finite Element Methods for Solving the Stokes Problem.- 1. Mixed Approximation of an Abstract Problem.- 1.1. A Mixed Variational Problem.- 1.2. Abstract Mixed Approximation.- 2. The "Stream Function-Vorticity-Pressure" Method for the Stokes Problem in Two Dimensions.- 2.1. A Mixed Formulation.- 2.2. Mixed Approximation and Application to Finite Elements of Degree l.- 2.3. The Technique of Mesh-Dependent Norms.- 3. Further Topics on the "Stream Function-Vorticity-Pressure" Scheme.- 3.1. Refinement of the Error Analysis.- 3.2. Super Convergence Using Quadrilateral Finite Elements of Degree l.- 4. A "Stream Function-Gradient of Velocity Tensor" Method in Two Dimensions.- 4.1. The Hellan-Herrmann-Johnson Formulation.- 4.2. Approximation with Triangular Finite Elements of Degree l.- 4.3. Additional Results for the Hellan-Herrmann-Johnson Scheme.- 4.4. Discontinuous Approximation of the Pressure.- 5. A "Vector Potential-Vorticity" Scheme in Three Dimensions.- 5.1. A Mixed Formulation of the Three-Dimensional Stokes Problem.- 5.2. Mixed Approximation in H(curl ?).- 5.3. A Family of Conforming Finite Elements in H(curl ?).- 5.4. Error Analysis for Finite Elements of Degree l.- 5.5. Discontinuous Approximation of the Pressure.- IV. Theory and Approximation of the Navier-Stokes Problem.- 1. A Class of Nonlinear Problems.- 2. Theory of the Steady-State Navier-Stokes Equations.- 2.1. The Dirichlet Problem in the Velocity-Pressure Formulation.- 2.2. The Stream Function Formulation of the Homogeneous Problem..- 3. Approximation of Branches of Nonsingular Solutions.- 3.1. An Abstract Framework.- 3.2. Approximation of Branches of Nonsingular Solutions.- 3.3. Application to a Class of Nonlinear Problems.- 3.4. Non-Differentiable Approximation of Branches of Nonsingular Solutions.- 4. Numerical Analysis of Centered Finite Element Schemes.- 4.1. Formulation in Primitive Variables: Methods Using Discontinuous Pressures.- 4.2. Formulation in Primitive Variables: the Case of Continuous Pressures.- 4.3. Mixed Incompressible Methods: the "Stream Function-Vorticity" Formulation.- 4.4. Remarks on the "Stream Function-Gradient of Velocity Tensor" Scheme.- 5. Numerical Analysis of Upwind Schemes.- 5.1. Upwinding in the Stream Function-Vorticity Scheme.- 5.2. Error Analysis of the Upwind Scheme.- 5.3. Approximating the Pressure with the Upwind Scheme.- 6. Numerical Algorithms.- 2.11. General Methods of Descent and Application to Gradient Methods.- 2.12. Least-Squares and Gradient Methods to Solve the Navier-Stokes Equations.- 2.13. Newton's Method and the Continuation Method.- References.- Index of Mathematical Symbols.

The instability of barotropic circular vortices

Abstract: The linear, normal mode instability of barotropic circular vortices with zero circulation is examined in the f-plane quasigeostrophic equations. Equivalents of Rayleigh's and Fjortoft's criteria and the semicircle theorem for parallel shear flow are given, and the energy equation shows the instability to be barotropic. A new result is that the fastest growing perturbation is often an internal instability, having a finite vertical scale, but may also be an external instability, having no vertical structure. For parallel shear flow the fastest growing perturbation is always an external instability; this is Squire's theorem. Whether the fastest growing perturbation is internal or external depends upon the profile: for mean flow streamfunction profiles which monotonically decrease with radius, the instability is internal for less steep profiles with a broad velocity extremum and external for steep profiles with a narrow velocity extremum. Finite amplitude, numerical model calculations show that this ...

The force on an axisymmetric body in linearized, time-dependent motion: a new memory term

Abstract: In contrast to the steady Stokes equations for creeping motion, the time-dependent linearized Navier–Stokes equations have only been solved for very restricted geometries, the solution for the sphere being the sole solution for an isolated finite body. In the present paper, the linearized Navier–Stokes equations are further explored and a simple expression is derived which relates the force on an arbitrary axisymmetric body in oscillatory motion to the solution for the stream function in the far field. This result is applied to the case of a slightly eccentric spheroid and it is shown that the total hydrodynamic force contains four terms, three of which correspond to the classical solutions for the Stokes drag, added mass and Basset force on the perturbed sphere; the fourth term is only present when the body is non-spherical. In contrast to the three classical forces, the new term is not a simple power of the dimensionless frequency parameter iL2ω/ν, in which L is a length-scale, ω is the frequency of oscillation and ν is the kinematic viscosity of the fluid. A Laplace superposition is then used to find the force on the spheroid in an arbitrary axisymmetric motion with velocity U(t). The new memory term decays faster than the Basset force at large times and is bounded at short times.

A pseudo-spectral solution of vorticity-stream function equations using the influence matrix technique

Abstract: A method for computing unsteady two dimensional incompressible flows, using the vorticity and the stream function as dependent variables approximated by Chebyshev polynomial expansions is presented. Boundary conditions for the vorticity are derived by the influence matrix technique. The theoretical and numerical difficulties associated with the two-dimensional formulation of the method are discussed. Numerical results illustrate the properties of the method.

Turbulent flow in stirred tanks. Part II: A two‐scale model of turbulence

Abstract: The two-scale turbulence concept is recommended for modeling the turbulence in a baffled vessel equipped with a Rushton-type turbine impeller. A three-equation isotropic turbulence model is proposed that employs the balance equations for: the kinetic energy of the large scale vortices; the kinetic energy of the inertial subrange eddies; and the dissipation rate of the small-scale turbulence. The energy transfer rate from the large-scale vortices is prescribed algebraically. Flow patterns are modeled by solving the transport equations for vorticity, stream function, and tangential momentum. The Reynolds stresses are modeled by means of the effective viscosity, based on the three-equation model of turbulence. The calculated profiles of the mean velocity at the tank wall agree with experimental data obtained in the same system by means of a Pitot tube.

Gas Flow in Nozzles

Abstract: 1 Introduction- 2 General Theory of Nozzle Gas Flows- 21 Basic Equations and Problem Formulation- 211 Basic Equations- 212 Characteristics- 213 The Direct Problem- 214 The Inverse Problem- 22 Some Elementary Theories- 221 One-Dimensional Theory- 222 Radial Flows or Source/Sink Flows- 223 Prandtl-Meyer Flow- 23 Variational Problems of Gas Dynamics of Internal Flows- 3 Numerical Methods of Studying Nozzle Gas Flows- 31 Methods of Solving Relaxation Equations- 32 Methods of Calculating Plane and Axisymmetric Supersonic Flows- 321 Method of Characteristics- 322 Shock-Smearing Methods- 33 Methods of Calculating Three-Dimensional Supersonic Flows- 331 Method of Characteristics- 332 Difference Methods- 34 Methods of Solving the Inverse Problem of the Theory of Nozzles- 35 Methods of Solving the Direct Problem of the Theory of Nozzles- 4 Asymptotic Methods in the Theory of Nozzles- 41 Source-Sink Method and Inverse Problem for Incompressible Fluid- 411 Plane Flow- 412 Axisymmetric Flow- 413 Three-Dimensional Flow- 414 Solution of the Inverse Problem for Incompressible Fluid- 42 Expansion in a Stream Function Series- 421 Three-Dimensional Flow- 422 Plane and Axisymmetric Flows- 423 Two-Phase Flows- 43 Asymptotic Methods in the Transonic Region- 431 Method of Small Perturbations- 432 Series Expansion in the Vicinity of a Rectilinear Sonic line- 44 Solution in the Neighbourhood of the Infinite Point in the Subsonic Region- 45 Method of Small Perturbations for Flows Close to Radial- 5 Nozzles of Jet Engines- 51 Flow Peculiarities in the Subsonic and Transonic Parts of a Nozzle- 511 Nozzle with a Rectilinear Surface of Transition- 512 Nozzle with a Curvilinear Surface of Transition- 513 Local Deceleration Zones- 514 Optimum Shape of the Subsonic Part of a Nozzle- 515 Flow Rate Coefficient and Critical Pressure Drop in Nozzles with Curvilinear Transition Surfaces- 516 Profiling of the Acceleration Part of a Jet Engine Nozzle- 52 Profiling of Supersonic Parts of Jet Engine Nozzles and Impulse Losses- 521 Profiling of Supersonic Parts of Jet Engine Nozzles- 522 Impulse Losses- 523 Thrust Changes Caused by Deviations from the Design Flow Regime- 524 Numerical and Experimental Simulation of Flows in Jet Engine Nozzles- 53 Main Principles of Choosing a Jet Engine Nozzle- 6 Flows with Physico-Chemical Transformations- 61 Isentropic Flows and Intermolecular Interaction- 611 Isentropic Flows- 612 Intermolecular Interaction- 62 Flows with Nonequilibrium Chemical Reactions- 621 Basic Equations, Method of Prediction and the One-Dimensional Approximation- 622 Major Characteristic Features of Chemical Nonequilibrium Flows- 623 Plane and Axisymmetric Flows- 624 Approximate Methods of Calculating Nonequilibrium Flows- 63 Flows with Vibrational Relaxation- 631 Relaxation Equations and Methods of Prediction- 632 Results of Calculations in the One-Dimensional Approximation- 633 Plane and Axisymmetric Flows- 64 Two-Phase Flows- 641 Basic Concepts: One-Dimensional Approximation, Equilibrium and Frozen Flows- 642 One-Dimensional Nonequilibrium Flows Without Phase Transitions Impulse Losses- 643 One-Dimensional Flows with Interacting Particles- 644 One-Dimensional Flows with Phase Transitions- 645 Flows in Axisymmetric and Plane Nozzles- 65 Conductive Gas Nozzle Flows in the Presence of an Electromagnetic Field- 66 Multilayer Nozzle Flows- 661 One-Dimensional Approximation- 662 Axisymmetric Flows- 7 Special Nozzles, Three-Dimensional Flows, Viscosity Effect- 71 Annular Nozzles and Wind Tunnel Nozzles- 711 Some Schemes of Annular Nozzles and Methods of Calculation- 712 Off-Design Flow Regimes in Annular Nozzles- 713 Wind Tunnel Nozzles- 72 Conical Nozzles- 73 Swirling Nozzle Flows- 731 Introductory Notes- 732 Radially Balanced Flows- 733 Axisymmetric Swirling Flows- 74 Three-Dimensional Nozzle Flows- 741 Some Results of Analytical and Experimental Studies- 742 Method of Small Perturbations and Determination of Side Forces and Moments- 743 Numerical Investigation of Three-Dimensional Nozzle Gas Flows- 744 Experimental Studies of Side Forces and Moments- 75 Flows at Small Reynolds Numbers- Nomenclature- References

Simulation of reacting flow during filling in reaction injection molding (RIM)

Abstract: This paper deals with computer simulation of the filling stage of the Reaction Injection Molding (RIM) process for cavities of rectangular, cylindrical, and disc shapes. The computer model is in two parts: the main flow and the flow by the moving front. In the main flow part, the transient equations of axial momentum, energy and species conservation and also the continuity equation are solved numerically by finite-difference methods using a moving, changing mesh. In the flow front part, which is quite novel, the transient (parabolic) vorticity, energy and species conservation equations and the elliptic stream function equation are again solved by finite-difference methods. Results are presented for all three cavity shapes and those for rectangular cavities are compared with the experimental results of previous investigators.

An Objective Analysis of the POLYMODE Local Dynamics Experiment. Part II: Streamfunction and Potential Vorticity Fields during the Intensive Period

Abstract: An analysis is made of the geostrophic streamfunction, potential vorticity and dynamical balances for the mesoscale flow during the intensive Period of the POLYMODE Local Dynamics Experiment The methodology is Objective analysis based upon three-dimensional, anisotropic covariance functions and an expansion in the vertical modes of linear theory. The flow field during the Intensive Period is highly anisotropic in both vertical modes. The barotropic mode behaves as a propagating wave with significant rotation in the latter part of the period; its dynamical balances are substantially linear except at the time of the rotation, when there is also a transfer of energy from the first baroclinic mode. The baroclinic flow exhibits a fonotogenetic intensification into a strong jet due to straining of an initially large-scale baroclinic flow by the barotropic wave, together with some aspects of phase propagation as well. The baroclinic modal dynamical balance has identifiable linear contributions, but they...

On the nonlinear critical layer below a nonlinear unsteady surface wave

Abstract: A theory is presented for the development of the nonlinear critical layer below an unsteady free surface wave, of amplitude e, described by the Korteweg-de Vries (KdV) equation. The problem is formulated (via the Euler equations) for wave propagation over an arbitrary shear in a two-dimensional channel which contains a critical level. The equations are scaled so as to be valid in the far-field regime of the surface wave, appropriate to the existence of the KdV equation, i.e. long waves. The regions above and below the critical layer are solved (to O(e), as e → 0) and thence expanded in the neighbourhood of the critical layer itself. The symmetry of the critical layer solution, assuming that it exists, is then sufficient to determine the Burns integral for the linearized wave speed, and the relevant KdV equation. These turn out to be the classical results evaluated in terms of finite parts.The critical layer, of thickness O(e½), is analysed to O(e) and matched to the outer regions of the flow field. The initial configuration is taken to contain no closed streamlines, and so the vorticity can, presumably, be assigned from the undisturbed conditions at infinity. The initial surface profile must therefore contain a single peak, but by virtue of the KdV equation this can evolve into any number of solitons. Between consecutive pairs of peaks there will now appear regions of closed streamlines (cat's-eyes) with known vorticity. No recourse to a viscous argument is necessary to uniquely determine this vorticity. However, it is shown that the vorticity cannot be prescribed arbitrarily at all orders, initially: the long-wave assumption imposes a certain structure on the problem, and then the continuity of stream function and particle velocity fixes the vorticity. This agrees with the work of Varley & Blyth (1983) on the hydraulic equations. The vorticity inside the separating streamlines is obtained to O(e), but it is shown that for unsteady motion this asymptotic expansion is not uniformly valid as the bounding streamlines are approached. An alternative method, which exploits Varley & Blythe's approach, is used to confirm the correctness of our results away from these boundaries, and to indicate that a non-uniformity is present near the separating streamlines. Thus the model requires the inclusion of a vortex sheet; for steady flow a jump in vorticity is sufficient. The removal of the discontinuity by allowing a distortion of the main flow outside the critical layer is briefly discussed.Some results are presented for the formation of a single cat's-eye by using the exact 2-soliton solution of the KdV equation.

Computation of the Streamfunction and Velocity Potential and Reconstruction of the Wind Field

Abstract: A wind field given over a limited domain can be partitioned into nondivergent and irrotational components in an infinity of ways. A particular solution, selected by requiring the velocity potential to vanish on the boundary, has minimum divergent kinetic energy and is numerically easy to obtain. The reconstruction of the wind field from the vorticity and divergence together with the boundary velocity is more difficult, since the potential equations are coupled by the boundary conditions. A numerical procedure is devised, which solves the two potential equations simultaneously, modifying both interior and boundary values in a converging iterative technique. The method is capable of reconstructing the wind field to any accuracy desired.

Equal‐energy streamlines

Abstract: Energy streamlines show the direction of energy flow in a sound field: The intensity vector at any point is tangential to the streamline passing through that point. In earlier work, a procedure for computing energy streamlines was described [J. Acoust. Soc. Am. 78, 758–762 (1985)], but it did not give equally spaced streamlines. The latter have certain advantages, and a method for computing them is presented here. It is shown that in sound fields which vary in only two space dimensions, a scalar stream function exists, the contours of which are the energy streamlines. The stream function is obtained from the intensity function (or values) by an integration procedure. Examples of equally spaced streamlines are given for the point‐driven, fluid‐loaded plate, and for the baffled rigid piston.

Stream function and stream-function-coordinate (SFC) formulation for inviscid flow field calculations

Abstract: Three-dimensional, steady, inviscid, compressible, isoenergetic, rotational flows can be completely described by two families of stream functions. In the present study, explicit forms of the governing equations for these two stream functions of baratropic fluid flows are derived. With the assumptions of two-dimensional or axisymmetric, incompressible and/or irrotational flows, this formulation can be reduced to the familiar special cases. The concept of stream-function-coordinate (SFC) is introduced along with some relevant examples. Exact gas dynamic equations using stream functions as independent variables are also presented. Applications and difficulties involved with the SFC concept are clearly exposed, and several possible ways to resolve these problems are discussed.

A computer simulation of transient electrohydrodynamic motion in stressed dielectric liquids

Abstract: Using a finite difference formulation of the Navier-Stokes equation coupled with a mathematical description of electrical charge migration. The authors examined transient electrohydrodynamic (EHD) phenomena in an electrically stressed liquid duct. Perturbations resulting from electrode injection are studied as a function of time by examining the liquid vorticity vector and stream function. The response of local pressure to advancing charge fronts is evaluated. The theoretical simulation is supported by previous laboratory studies of EHD motion. Experimentally recorded flow patterns are compared with the predicted results.

A Viscous-Inviscid Interaction Procedure—Part 1: Method for Computing Two-Dimensional Incompressible Separated Channel Flows

Abstract: A viscous-inviscid interaction scheme has been developed for computing steady incompressible laminar and turbulent flows in two-dimensional duct expansions. The viscous flow solutions are obtained by solving the boundary-layer equations inversely in a coupled manner by a finite-difference scheme; the inviscid flow is computed by numerically solving the Laplace equation for stream function using an ADI finite difference procedure. The viscous and inviscid solutions are matched iteratively along displacement surfaces. Details of the procedure are presented along with example applications to separated flows. The results compare favourably with experimental data. Applications to turbulent flows over a rearward-facing step are described in part 2. See next Abstract.

Traveling‐wave solutions and the coupled Korteweg–de Vries equation

Abstract: Some coupled nonlinear equations are considered for studying traveling‐wave solutions. By introducing a stream function Ψ it is shown that if one of the solutions is of the form v≡v(x−ct), the other also must be of the form u≡u(x−ct). In addition, the possibility of including cubic nonlinear terms has been considered and such a system, assuming that the solutions are of the traveling‐wave type, has been solved.

A method for the simulation of plane or axisymmetric flows of incompressible fluids based on the concept of the stream function

Abstract: A new numerical method is presented for the simulation of flows of incompressible fluids in plane or axisymmetric flows. Under certain assumptions, the physical domain can be transformed into a rectangular domain. This method can involve free surface flow problems. In Newtonian and non‐Newtonian cases, the relevant equations are non‐linear and the solution is carried out in the transformed domain where the stream lines are parallel straight lines.

Free Surface Stokes Flow Over an Obstacle

Abstract: The Stokes flow of a viscous fluid layer over a cylindrical obstacle on a tilted plane is determined by means of an integral equation method. The method is based on an integral formula for the stream function expressed in terms of two field quantities on the surface of the obstacle and the fluid velocity in the free surface. From the formula three integral equations are derived. The location of the surface and the field quantities are found by means of an iterative procedure. Numerical examples with fluids with or without surface tension are presented.

Solving problems in fluid mechanics

Abstract: 1 Dimensional Analysis 2 Dynamical Similarity Problems 3 Vortex Motion And Radial Flow 4 Streamlines And Stream Function 5 Gases At Rest 6 Flow Of Gases 7 Viscous Flow 8 Turbulent Flow 9 Flow Round Totally Immersed Bodies 10 Waterhammer And Pressure Transients 11 Non-Uniform Flow In Channels 12 Impulse And Reaction Turbines 13 Centrifugal Pumps 14 Reciprocating Pumps 15 Computational Fluid Dynamics Index

An Inverse (Design) Problem Solution Method for the Blade Cascade Flow on Streamsurface of Revolution

Abstract: On the basis of the fundamental equations of aerothermodynamics a method for solving the inverse (design) problem of blade cascade flow on the blade-to-blade streamsurface of revolution is suggested in the present paper. For this kind of inverse problem the inlet and outlet flow angles, the aerothermodynamic parameters at the inlet, and the other constraint conditions are given. Two approaches are proposed in the present paper: the suction-pressure-surface alternative calculation method (SSAC) and the prescribed streamline method (PSLM). In the first method the metric tensor (blade channel width) is obtained by alternately fixing either the suction or pressure side and by revising the geometric form of the other side from one iteration to the next. The first step of the second method is to give the geometric form of one of the streamlines. The velocity distribution or the mass flow rate per unit area on that given streamline is estimated approximately by satisfying the blade thickness distribution requirement. The stream function in the blade cascade channel is calculated by assuming initial suction and pressure surfaces and solving the governing differential equations. Then, the distribution of metric tensor on the given streamline is specified by the stream function definition. It is evident that the square root of the metric tensor is a circumferential width of the blade cascade channel for the special nonorthogonal coordinate system adopted in the present paper. The iteration procedure for calculating the stream function is repeated until the convergence criterion of the metric tensor is reached. A comparison between the solutions with and without consideration of viscous effects is also made in the present paper.

Estimation of water particle velocities of shallow water waves by a modified transfer function method

Abstract: This paper Is intended to propose a simple, yet highly reliable approximate method which uses a modified transfer function in order to evaluate the water particle velocity of finite amplitude waves at shallow water depth in regular and irregular wave environments. Using Dean's stream function theory, the linear function is modified so as to include the nonlinear effect of finite amplitude wave. The approximate method proposed here employs the modified transfer function. Laboratory experiments have been carried out to examine the validity of the proposed method. The approximate method is shown to estimate well the experimental values, as accurately as Dean's stream function method, although its calculation procedure is much simpler than that of Dean's method.

Nonuniqueness in Stream Function Wave Theory

Abstract: The Stream Function Wave Theory has been shown to produce waves with multiple crests. This paper shows that the multiple-crested waves are not real, but an attempt by the numerical theory to represent a periodic wave train of shorter period. A procedure to restrict the solution technique to only single-crested waves is provided.

Orbital flow past a cylinder: a numerical approach

Abstract: Orbital flow past a cylinder is relevant to offshore structures. The numerical scheme presented here is based on a finite-difference solution of the Navier–Stokes equations. Alternating-directional-implicit (ADI) and successive-over-relaxation (SOR) techniques are used to solve the vorticity-transport and stream-function equations. Theoretical simulations to low Reynolds number flows (up to 1000) are discussed for cases involving uniform flow past stationary and rotating cylinders and orbital flow past a cylinder. The separation points for cylinders that are rotating or immersed in an orbital flow are deduced from velocity profiles through the boundary layer using a hybrid mesh scheme. During the initial development of orbital flow surface vorticity on the impulsively started cylinder dominates the flow. A vortex then detaches from behind the cylinder and establishes the flow pattern of the orbit. After some time a collection of vortices circles the orbit and distorts its shape a great deal. These vortices gradually spiral outward as others detach from the cylinder and join the orbital path.

Body of revolution comparisons for axial- and surface-singularity distributions

Abstract: GIVEN axisymmetric body immersed in an incompressible uniform flow can be represented by two different types of singularity methods. The simpler type is the method of axial singularities. This representation is by a distribution of line sources and sinks aligned with the uniform flow to form the closed body shape. The surface of the body is represented by a constant value of stream function, typically taken to be zero. The more complex, but far more versatile, technique is the method of surface-distr ibuted singularities which are typically ring sources and sinks, but can also be ring doublets or vortices. The surface-distribution method is more complex because the method of describing the body involves control points which are also singular. In the following discussion, a comparison of the surface velocities generated by the two different methods will be made. It will be shown that the simplicity of the axialsingularity method is actually its major problem in representing axisymmetric body shapes. It will also be evident that the complexity of the surface-singularity method is its strength in representing axisymmetric body shapes.