Modeling of Three-Dimensional Flow in Turning Channels

1984 ◽  
Vol 106 (3) ◽  
pp. 682-691 ◽  
Author(s):  
I. M. Khalil ◽  
H. G. Weber
Keyword(s):  
Stokes Equations ◽  
Reverse Flow ◽  
Three Dimensional ◽  
Solution Procedure ◽  
Three Dimensions ◽  
Navier Stokes ◽  
Pressure Gradients ◽  
Dean Number ◽  
Navier Stokes Equations ◽  
Curved Channels

The structure of developing flows inside curved channels has been investigated numerically using the time-averaged Navier Stokes equations in three dimensions. The equations are solved in primitive variables using finite difference techniques. The solution procedure involves a combination of repeated space-marching integration of the governing equations and correction for elliptic effects between two marching sweeps. Type-dependent differencing is used to permit downstream marching even in the reverse-flow regions. The procedure is shown to allow efficient calculations of turbulent flow inside strongly curved channels as well as laminar flow inside a moderately curved passage. Results obtained in both cases indicate that the flow structure is strongly controlled by local imbalance between centrifugal forces and pressure gradients. Furthermore, distortion of primary flow due to migration of low momentum fluid caused by secondary flow is found to be largely dependent on the Reynolds number and Dean number. Comparison with experimental data is also included.

A Calculation Scheme for Three-Dimensional Viscous Incompressible Flows

1987 ◽  
Vol 109 (4) ◽  
pp. 345-352 ◽  
Author(s):  
M. Reggio ◽  
R. Camarero
Keyword(s):  
Incompressible Flows ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Numerical Procedure ◽  
Numerical Data ◽  
Momentum Balance ◽  
Navier Stokes ◽  
Test Cases ◽  
Pressure Gradients ◽  
Navier Stokes Equations

A numerical procedure to solve three-dimensional incompressible flows in arbitrary shapes is presented. The conservative form of the primitive-variable formulation of the time-dependent Navier-Stokes equations written for a general curvilinear coordiante system is adopted. The numerical scheme is based on an overlapping grid combined with opposed differencing for mass and pressure gradients. The pressure and the velocity components are stored at the same location: the center of the computational cell which is used for both mass and the momentum balance. The resulting scheme is stable and no oscillations in the velocity or pressure fields are detected. The method is applied to test cases of ducting and the results are compared with experimental and numerical data.

Numerical Simulation of Asymmetric Exhaust Flows Using an Actuator Disc Blade Row Model

2002 ◽  
Author(s):  
Jianjun Liu
Keyword(s):  
Numerical Simulation ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Solution Procedure ◽  
Navier Stokes ◽  
Exhaust Hood ◽  
Navier Stokes Equations ◽  
Multigrid Algorithm ◽  
Blade Row ◽  
Actuator Disc

This paper describes the numerical simulation of the asymmetric exhaust flows by using a 3D viscous flow solver incorporating an actuator disc blade row model. The three dimensional Reynolds-Averaged Navier-Stokes equations are solved by using the TVD Lax-Wendroff scheme. The convergence to a steady state is speeded up by using the V-cycle multigrid algorithm. Turbulence eddy viscosity is estimated by the Baldwin-Lomax model. Multiblock method is applied to cope with the complicated physical domains. Actuator disc model is used to represent a turbine blade row and to achieve the required flow turning and entropy rise across the blade row. The solution procedure and the actuator disc boundary conditions are described. The stream traces in various sections of the exhaust hood are presented to demonstrate the complicity of the flow patterns existing in the exhaust hood.

Three-dimensional natural convection in a box: a numerical study

1977 ◽  
Vol 83 (1) ◽  
pp. 1-31 ◽  
Author(s):  
G. D. Mallinson ◽  
G. De Vahl Davis
Keyword(s):  
Natural Convection ◽  
Stokes Equations ◽  
Numerical Study ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Aspect Ratios ◽  
Inertial Interaction ◽  
Longitudinal Temperature

The solution of the steady-state Navier–Stokes equations in three dimensions has been obtained by a numerical method for the problem of natural convection in a rectangular cavity as a result of differential side heating. In the past, this problem has generally been treated as though it were two-dimensional. The solutions explore the three-dimensional motion generated by the presence of no-slip adiabatic end walls. For Ra = 104, the three-dimensional motion is shown to be the result of the inertial interaction of the rotating flow with the stationary walls together with a contribution arising from buoyancy forces generated by longitudinal temperature gradients. The inertial effect is inversely dependent on the Prandtl number, whereas the thermal effect is nearly constant. For higher values of Ra, multiple longitudinal flows develop which are a delicate function of Ra, Pr and the cavity aspect ratios.

Calculation of Turbulent Flows in a Hydraulic Turbine Draft Tube

1990 ◽  
Vol 112 (3) ◽  
pp. 257-263 ◽  
Author(s):  
M. Agouzoul ◽  
M. Reggio ◽  
R. Camarero
Keyword(s):  
Turbulent Flows ◽  
Stokes Equations ◽  
Draft Tube ◽  
Three Dimensional ◽  
Hydraulic Turbine ◽  
Navier Stokes ◽  
Pressure Gradients ◽  
Navier Stokes Equations ◽  
Conservative Form ◽  
Hydraulic Turbine Draft Tube

A numerical method to simulate three-dimensional incompressible turbulent flows has been developed and applied to the calculation of various flow situations in a draft tube. The conservative form of the primitive-variable formulation of the Reynolds averaged Navier-Stokes equations, written for a general curvilinear co-ordinate system, is employed. An overlapping grid combined with opposed differencing for mass and pressure gradients is used. All the properties are stored at the center of the same computational cell which is used for mass and transport balances. The k–ε model is used to describe the turbulent flow. The boundary conditions for the turbulent properties are treated with a particular attention.

A Simultaneous Variable Solution Procedure for Laminar and Turbulent Flows in Curved Channels and Bends

2000 ◽  
Vol 122 (3) ◽  
pp. 552-559 ◽  
Author(s):  
Jianrong Wang ◽  
Siamack A. Shirazi
Keyword(s):  
Experimental Data ◽  
Numerical Solution ◽  
Numerical Method ◽  
Solution Method ◽  
Stokes Equations ◽  
Turbulence Models ◽  
Solution Procedure ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Curved Channels

Direct Numerical Simulation of turbulent flow requires accurate numerical techniques for solving the Navier-Stokes equations. Therefore, the Navier-Stokes equations in general orthogonal and nonorthogonal coordinates were employed and a simultaneous variable solution method was extended to solve these general governing equations. The present numerical method can be used to accurately predict both laminar and turbulent flow in various curved channels and bends. To demonstrate the capability of this numerical method and to verify the method, the time-averaged Navier-Stokes equations were employed and several turbulence models were also implemented into the numerical solution procedure to predict flows with strong streamline curvature effects. The results from the present numerical solution procedure were compared with available experimental data for a 90 deg bend. All of the turbulence models implemented resulted in predicted velocity profiles which were in agreement with the trends of experimental data. This indicates that the solution method is a viable numerical method for calculating complex flows. [S0098-2202(00)01803-4]

Laminar Incompressible Viscous Flow in Curved Ducts of Regular Cross-Sections

1977 ◽  
Vol 99 (4) ◽  
pp. 640-648 ◽  
Author(s):  
K. N. Ghia ◽  
J. S. Sokhey
Keyword(s):  
Cross Sections ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Dean Number ◽  
Navier Stokes Equations ◽  
Flow Parameters ◽  
Incompressible Viscous Flow ◽  
Square Ducts ◽  
Curved Ducts

The laminar three-dimensional flow in curved ducts has been analyzed for an incompressible viscous fluid. The mathematical model is formulated using three-dimensional parabolized Navier-Stokes equations. The equations are generalized using two indices which permit the choice of Cartesian or cylindrical coordinate systems and straight or curved ducts. The solutions are obtained numerically using an ADI method for a number of duct geometries and flow parameters. The study presents detailed results for developing laminar flow in rectangular curved ducts; also, the effect of longitudinal curvature on secondary flow is fully analyzed. An investigation is made of the occurrence of Dean’s instability and, for curved square ducts, it is found to first appear at Dean number ≃ 143.

Exponential asymptotic stability of a class of dynamical systems with applications to models of turbulent flow in two and three dimensions

2012 ◽  
Vol 142 (2) ◽  
pp. 225-238 ◽  
Author(s):  
Joel Avrin
Keyword(s):  
Dynamical Systems ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Navier Stokes ◽  
Bounded Domains ◽  
First Eigenvalue ◽  
Navier Stokes Equations ◽  
Exponential Asymptotic Stability ◽  
Decaying Turbulence

We consider a class of dynamical systems of the form du/dt + Bu + F(u) = b on a Hilbert space H where the self-adjoint linear operator B is positive with a strictly positive first eigenvalue and b = b0 + b1 such that (b0, Bv) = 0 for all v ∈ H. Given two solutions u and v, we set u − v = w and show that if u(t) → 0 and v(t) → 0 as t → ∞, then in fact eventually w(t) → 0 at an exponential rate. We apply these results to the two-dimensional Navier–Stokes equations (NSEs), the three-dimensional hyperviscous NSEs and the three-dimensional NS-α equations on bounded domains and also establish stability in the sense of Lyapunov; for these systems we assume a condition on b1 to impose decaying turbulence. We also show for the case of decaying turbulence that Leray solutions of the three-dimensional NSEs on bounded domains eventually become regular in addition to decaying to zero. In particular, they eventually satisfy the conditions needed for the abstract stability results.

scholarly journals Large data existence theory for three-dimensional unsteady flows of rate-type viscoelastic fluids with stress diffusion

2020 ◽  
Vol 10 (1) ◽  
pp. 501-521 ◽  
Author(s):  
Michal Bathory ◽  
Miroslav Bulíček ◽  
Josef Málek
Keyword(s):  
Stokes Equations ◽  
Constitutive Relations ◽  
Three Dimensional ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Three Dimensions ◽  
Existence Theory ◽  
Navier Stokes ◽  
Time Interval ◽  
Navier Stokes Equations

Abstract We prove that there exists a weak solution to a system governing an unsteady flow of a viscoelastic fluid in three dimensions, for arbitrarily large time interval and data. The fluid is described by the incompressible Navier-Stokes equations for the velocity v, coupled with a diffusive variant of a combination of the Oldroyd-B and the Giesekus models for a tensor 𝔹. By a proper choice of the constitutive relations for the Helmholtz free energy (which, however, is non-standard in the current literature, despite the fact that this choice is well motivated from the point of view of physics) and for the energy dissipation, we are able to prove that 𝔹 enjoys the same regularity as v in the classical three-dimensional Navier-Stokes equations. This enables us to handle any kind of objective derivative of 𝔹, thus obtaining existence results for the class of diffusive Johnson-Segalman models as well. Moreover, using a suitable approximation scheme, we are able to show that 𝔹 remains positive definite if the initial datum was a positive definite matrix (in a pointwise sense). We also show how the model we are considering can be derived from basic balance equations and thermodynamical principles in a natural way.

Eddy viscosity of three-dimensional flow

1995 ◽  
Vol 288 ◽  
pp. 249-264 ◽  
Author(s):  
A. Wirth ◽  
S. Gama ◽  
U. Frisch
Keyword(s):  
Large Scale ◽  
Eddy Viscosity ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Cubic Symmetry ◽  
A Value ◽  
Spatially Periodic

Detailed theoretical and numerical results are presented for the eddy viscosity of three-dimensional forced spatially periodic incompressible flow.As shown by Dubrulle & Frisch (1991), the eddy viscosity, which is in general a fourth-order anisotropic tensor, is expressible in terms of the solution of auxiliary problems. These are, essentially, three-dimensional linearized Navier–Stokes equations which must be solved numerically.The dynamics of weak large-scale perturbations of wavevector k is determined by the eigenvalues – called here ‘eddy viscosities’ – of a two by two matrix, obtained by contracting the eddy viscosity tensor with two k-vectors and projecting onto the plane transverse to k to ensure incompressibility. As a consequence, eddy viscosities in three dimensions, but not in two, can become complex. It is shown that this is ruled out for flow with cubic symmetry, the eddy viscosities of which may, however, become negative.An instance is the equilateral ABC-flow (A = B = C = 1). When the wavevector k is in any of the three coordinate planes, at least one of the eddy viscosities becomes negative for R = 1/v > Rc [bsime ] 1.92. This leads to a large-scale instability occurring for a value of the Reynolds number about seven times smaller than instabilities having the same spatial periodicity as the basic flow.

Effects of stator splitter blades on aerodynamic performance of a single-stage transonic axial compressor

2020 ◽  
Vol 14 (4) ◽  
pp. 7369-7378
Author(s):  
Ky-Quang Pham ◽  
Xuan-Truong Le ◽  
Cong-Truong Dinh
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Axial Compressor ◽  
Aerodynamic Performance ◽  
Numerical Results ◽  
Single Stage ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Operating Stability ◽  
Length Coverage

Splitter blades located between stator blades in a single-stage axial compressor were proposed and investigated in this work to find their effects on aerodynamic performance and operating stability. Aerodynamic performance of the compressor was evaluated using three-dimensional Reynolds-averaged Navier-Stokes equations using the k-e turbulence model with a scalable wall function. The numerical results for the typical performance parameters without stator splitter blades were validated in comparison with experimental data. The numerical results of a parametric study using four geometric parameters (chord length, coverage angle, height and position) of the stator splitter blades showed that the operational stability of the single-stage axial compressor enhances remarkably using the stator splitter blades. The splitters were effective in suppressing flow separation in the stator domain of the compressor at near-stall condition which affects considerably the aerodynamic performance of the compressor.

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