Computational Fluid Dynamics (CFD) Since the 1950s, with the development of computers, a cross discipline between mathematics, fluid mechanics and computers has emerged. Its main research content is to solve the control equations of fluid mechanics through computer and numerical methods, and simulate and analyze the problems of fluid mechanics.
Fluid mechanics, like other disciplines, is developed through theoretical analysis and experimental research.Theoretical hydrodynamics and experimental hydrodynamics have long been two branches.Theoretical analysis is the quantitative result of solving problems with mathematical methods.However, there are only a few problems that can be solved by this method. Computational fluid dynamics is developed to make up for the shortcomings of analytical methods.
As early as the beginning of the 20th century, Richard had proposed the idea of using numerical methods to solve hydrodynamic problems.However, due to the complexity of the problem itself and the backwardness of computing tools at that time, this idea did not attract people's attention.Since the advent of electronic computers in the mid-1940s, it has become a reality to use electronic computers for numerical simulation and calculation.In 1963, FH. Harlow and JE. Using the IBM 7090 computer at that time, Fromm successfully solved the problem of the flow around a two-dimensional rectangular cylinder and gave the formation and evolution process of the wake vortex street, which attracted widespread attention.In 1965, Harlow and Fromm published the article "Computer Experiments in Fluid Dynamics", which made a remarkable introduction to the great role of computers in fluid mechanics.Since then, the mid-1960s has been regarded as a sign of the rise of computational fluid dynamics.
Although the history of computational fluid mechanics is not long, it has been widely used in various fields of fluid mechanics, and various numerical solutions have been formed accordingly.As far as the current situation is concerned, the main methods are finite difference method and finite element method.The finite difference method has been widely used in fluid mechanics.The finite element method is developed from solving the problems of solid mechanics.In recent years, there have been many applications in dealing with low-speed fluid problems, and they are still developing rapidly.
basic equation
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In order to illustrate the main methods of computational fluid dynamics, it is necessary to first understand the nature and classification of the basic equations of fluid dynamics.The basic equation of fluid mechanics was formulated by C-50. - M. - H. Navi and GG. It is called Navier Stokes equation, or N-S equation for short[1], the N-S equation of two-dimensional unsteady incompressible fluid is:
formula
Where u and v are velocity components along the x and y directions;T is the time;P is the pressure;ρ is the density;ν is the kinematic viscosity coefficient.Under different conditions, the mathematical properties of N-S equation are different.
① N-S equation describes the unsteady motion of viscous fluid with time.The time term and the higher-order derivative term on the right side of the equation determine the properties of the equation.It is similar to the two-dimensional heat conduction equation and belongs to the parabolic equation.
② The steady motion of viscous fluid omits the time term in the original equation.At this time, the nature of N-S equation depends on its high-order derivative term. Like Laplace equation, it is an elliptic equation.
③ The Euler equation of inviscid flow is obtained by omitting the viscous term on the right side of the N-S equation.It also applies to compressible fluids.It is not easy to judge the properties of Euler equation formally.Since most inviscid flows are irrotational, if the Euler equation is replaced by the velocity potential ψ, the equation of two-dimensional steady compressible flow is:
formula
Where c is the speed of sound.This formula is a second-order partial differential equation
formula
The general form of Btwo-AC
0.In the supersonic zone, Btwo-AC
0, i.e
, the above formula is similar to the wave equation, which is hyperbolic;In the subsonic region, Btwo-AC
0, i.e
The above formula is the same as the Laplace equation, which is elliptic.In a word, the equations of motion of fluid mechanics are extremely complex nonlinear partial differential equations, which have various types and are often mixed.To fully describe the motion of fluid, other equations must be considered at the same time, such as continuity equation, energy equation and state equation.Therefore, computational fluid dynamics (CFD), to a large extent, is to adopt and develop corresponding numerical solution methods for partial differential equations with different properties.
Numerical solution of low speed inviscid flow
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Numerical solution of low-speed inviscid flow under irrotational condition, the velocity potential of low-speed flow satisfies laplace equation or poisson equation.Many plane problems can be solved analytically by using complex function and conformal mapping, which is an important content of classical fluid mechanics.However, for objects with complex geometry, the following numerical solution must be used.
iterative method
This is a method of solving simultaneous equations by means of gradual approximation, and it is also the main numerical solution of elliptic differential equations.The program of this method is simple, and the storage and operation amount are relatively small. Generally, a group of initial values are assumed first, and then the new values on each node are calculated.Taking the five point format as an example, the new value on the dot is the average of the initial values of the adjacent four points.After the new value is calculated, the old value should be retained to calculate the new value of other points.This simple iteration converges slowly and is rarely used now.However, if the new value is slightly improved, the old value is flushed out with the calculated new value, and a relaxation factor is introduced to accelerate the convergence. The weighted average of the calculated new value and the original old value becomes the successive over relaxation method developed in the 1950s.
Time correlation method
This is a method of solving steady problems with unsteady equations, which is often used to solve N-S equations and Euler equations.Although the unsteady equation is used, the solution is not unsteady.According to the given initial conditions and constraints that change with time, the unsteady problem is to study the evolution process of flow with time.This kind of unsteady behavior is closely related to the given initial value.However, the initial value of the time correlation method is chosen randomly in principle, but it must meet the boundary conditions specified by the steady problem.In the process of solution, the change of flow with time does not represent a real physical process.When the time is long enough, the unknown function value is gradually independent of time, and then asymptotically tends to the steady solution.Therefore, the time correlation method is actually an iterative method, and the time variable is just used to record the number of iterations.
Alternating direction implicit method
The application of fluid mechanics is often a two-dimensional and three-dimensional space problem.Due to the requirement of stability, the time step is limited by the dimension. The higher the dimension is, the smaller the time step is required and the greater the calculation workload is.D. in the mid 1950sW. Peisman and J. Douglas proposed the so-called alternating direction implicit method to speed up the calculation.For example, in the two-dimensional unsteady equation, the first step is toxThe derivative ofyThe derivative of the direction uses the value of the previous one.Step 2yThe derivative of,xThe derivative of direction is the value calculated in the first step.The advantages of this method are good stability, sufficient second-order accuracy, and the difference equation generated is a tridiagonal matrix equation, which is easy to solve.
Finite fundamental solution
A numerical method for solving potential flow.The design of low-speed aircraft in the aviation industry uses potential theory to calculate various aerodynamic parameters, which is to solve two-dimensional or three-dimensional Laplace equations.In classical fluid mechanics, it is very successful to solve Laplace equation by superposition of fundamental solutions.The main point of this method is to replace the influence of wings and fuselage on the flow field with the distribution of sources, sinks and dipoles.Their strength is determined by the boundary conditions, and the results need to solve the integral equation.It can be solved for some simple cases, but it is difficult for general cases.The emergence of high-speed electronic computers has made a breakthrough in the numerical solution of this integral equation.Its main idea is to discretize the integral equation, which represents the sum of the continuous distribution of singular points such as sources and sinks in space.For example, if the wing and fuselage surfaces are divided into several small units, the intensity of singular points on each unit is averaged.Add up the sum of these singular points to get the total effect of the flow field.Therefore, it uses the summation of finite terms to replace integration, and the final solution is a set of algebraic equations.Since the fundamental solutions are all functions with singular points, this method is also called finite singular point method or scale method.(See finite basic solution)
Numerical solution of transonic flow
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The flow field of transonic flow is a mixed flow field with both subsonic and supersonic regions.Without considering the viscous effect and small disturbance, the steady two-dimensional velocity potential equation is of mixed type, that isVφxx+φyy=0, whereVIs the incoming flow Mach number ※ andφxIs a complex function of.V>0 is the subsonic region (elliptical), andV<0 is the supersonic zone (hyperbolic type).E. of the United StatesM. Muman and JD. Cole first used the mixed difference scheme in 1971, and successfully solved the steady small disturbance velocity potential equation by using the relaxation method.The hybrid difference scheme is to use the central difference scheme in the subsonic region, and the conditions on all adjacent nodes will affect the calculation points, while in the supersonic region, the upwind scheme is used, because the upstream upwind node is just the dependency area of the hyperbolic wave equation.(See numerical calculation of transonic flow)
Numerical solution of supersonic flow
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In supersonic flow, the main problem is how to deal with shock waves.There are two methods to deal with shock waves in supersonic flow field by numerical method.One is the method of arousing capture, and the other is the method of arousing assembly.The shock wave capture method does not need any special treatment for the shock wave itself, but directly or indirectly introduces the "viscous" term into the calculation formula to automatically calculate the position and strength of the shock wave in order to "capture" the shock wave.There are also two methods called artificial viscosity and format viscosity.The artificial viscosity method is J. von Neumann and RD. Richtmayer first proposed it in 1950. It is an automatic shock wave approximation method based on the physical theory of real viscous fluid.In this method, a viscous term is artificially added into the shock layer to make the shock discontinuity become a smooth transition region.In recent years, it has been widely used in supersonic flow.The viscous scheme is an indirect introduction of the viscous Lax scheme through some difference scheme.The Lax Wendhoff format and the McMark format have similar effects.The shock wave assembly method treats the shock wave as a discontinuity, and the shock wave jumping conditions must be met before and after the shock wave.But in ordinary coordinates, its implementation is very difficult.Generally, coordinate transformation is used to make the shock wave position (unknown at this time) coincide with a coordinate axis, and then the shock wave is regarded as the internal boundary.This kind of processing is more accurate, but it is also very troublesome and inconvenient to rent.The best way is to combine the shock wave capture method with the shock wave assembly method.For example, the shock assembly method is used for the detached shock wave outside the flow field, and the shock wave capture method is used for the shock wave inside the flow field.(See numerical solution of supersonic inviscid flow)
See Navier Stokes equation numerical solution, boundary layer equation numerical solution and turbulent numerical calculation for viscous flow work numerical solution.[1]
