Navier–Stokes equations

The Navier -Stokes equations [ navje stəʊks ː ] ( after Claude Louis Marie Henri Navier and George Gabriel Stokes ) describe the flow of Newtonian liquids and gases. The equations are therefore an extension of the Euler equation by the internal friction or viscosity.

In a narrower sense, especially in physics, is meant by the Navier -Stokes equations, the momentum equation for flows. In a broader sense, especially in computational fluid dynamics, this momentum equation is extended by the continuity equation and the energy equation, and then forms a system of nonlinear partial differential equations of second order. This is the basic mathematical model of fluid mechanics. In particular, the equations form from turbulence and boundary layers.

• 3.1 Theoretical solution
• 3.2 Numerical Solution
• 3.3 Calculation of turbulent flows

• 4.1 Euler equations
• 4.2 Stokes equation
• 4.3 Boussinesq approximation

History

1755 Leonhard Euler led forth the Euler equations that are used to describe the behavior of inviscid fluids. Prerequisite for this was his still valid definition of pressure in a fluid.

Stokes and Navier formulated independently in the first half of the 19th century (1827 or 1845), then the momentum equation for frictional Newtonian fluids, such as water, air or oil in differential form. 1843 published Barré de Saint- Venant a correct derivation of the Navier -Stokes equations, two years before Stokes did this, she sat down, however, the name of the Navier -Stokes equations by. Siméon Denis Poisson Also, she released 1831st An important advance in the theoretical and practical understanding delivered Ludwig Prandtl in 1904 with his boundary layer theory.

Formulation

Momentum equation

The Navier -Stokes equation in the strict sense, the pulse rate as an application of Newtonian axioms for a continuum. A commonly used form for compressible fluids is:

Here, the density, the pressure and the velocity of a particle in the flow. The vector describing the volume power density such as the gravity or the Coriolis force based on the unit volume, and has the unit newton / cubic meter. In the material constants, and is the dynamic viscosity and the first Lamé constant. In the literature they are often referred to as Lamé viscosity constants. Your determination is usually done on the experimentally well validated Stokes relation.

Another notation for the form used in the literature is:

Here, a resistance coefficient is defined that due to the condition that the trace of the deviatoric stress tensor must be zero, is related to the dynamic viscosity and the first Lamé constants in the following equation:

With the continuity equation and applying the Stokes' relation thereof is the equation for the pulse density:

To complete the equations of conservation of mass have to set ( the continuity equation) and conservation of energy are added. Depending on other assumptions are made on the fluid, the complete system results in different forms. The most commonly used are the incompressible Navier -Stokes equations.

Navier -Stokes equations for incompressible fluid

If the density along particle trajectories does not change, ie the flow is incompressible. This is, for example, a reasonable assumption for water. The continuity equation is simplified to the vanishing divergence of the velocity field

Which provides an alternative characterization of incompressible fluids. The momentum equation simplifies to:

Here, and are each for the physical pressure or the volume force based on the unit volume. is the dynamic viscosity. For an incompressible flow is completely described by a partial differential equation system of two equations for the two variables velocity and pressure as a function of time and place. The conservation of energy is not needed to close the system. This set of equations is also called incompressible Navier-Stokes equations with variable density. Application examples for this equation are problems of oceanography, when water of different salinity is indeed incompressible, but not a constant density.

In many practical problems, the flow is not only incompressible, but even has a constant density. Here you can divide by the density and include them in the differential operators:

In this equation, represents the ratio of physical pressure and density is the quotient of the volume force and density. Both quantities thus describe the pressure and the volume force related to the unit mass. The size is called kinematic viscosity and describes the diffusive transport of momentum.

The latter equations are usually called the incompressible Navier -Stokes equations, or referred to simply as the Navier -Stokes equations, because they are the most studied and used in practice most frequently in the literature. They apply to many important current issues, such as air currents far below the speed of sound ( Mach number <0.3 ), for water currents as well as for liquid metals. However, once change the densities of the considered fluids strong, such as in supersonic flows or in meteorology, provide the incompressible Navier -Stokes equations no suitable model of reality more is and must be replaced by the full ( compressible ) Navier -Stokes equations be.

Navier-Stokes equations for compressible fluids

This form of the Navier -Stokes equations is considered for a general ideal gas and consists of the equations for mass conservation, momentum conservation, energy conservation and the equation of state. Assuming that the density is constant along the particle trajectories, it is returned to the equations for incompressible fluids. A Entdimensionalisierung supplies various dimensionless parameters such as the Reynolds number or Prandtl number.

The notation: the derivative of size of the time divergence (or gradient ), the three position coordinates.

Conservation of mass

The mass conservation is here formulated with the momentum density, ie it arises

Conservation of momentum

The conservation of momentum reads in index notation

Where the Kronecker delta, and

The viscous stress tensor describes. Here, the dynamic viscosity and the i- th component of the gravity vector. In the alternative koordinateninvarianten notation, the equation of conservation of momentum

In which

The viscous stress tensor and the dynamic viscosity. is the stress tensor and is the unit tensor.

Energy conservation

The equation for conservation of energy is

In which

The enthalpy per unit mass. The heat flow, by means of the coefficient of thermal conductivity as

Be written. is a source term that describes, for example, the absorption and emission of greenhouse gases. The total energy per unit mass is the sum of internal (), kinetic and potential energy, it can be ( with height) that is written as

Equation of state

So we have four equations for five variables and the system is completed by the equation of state:

The thermodynamic quantities, density, pressure and temperature are linked by the ideal gas law:

Often you will additionally from a perfect gas with constant specific heat capacity. Then the integral and it is simplified:

In both cases, the adiabatic exponent of the gas constant and the specific heat sag coefficients for constant pressure and constant volume, respectively, through combined.

Boundary conditions

An important point in the Navier -Stokes equations is the experimentally well proven slip condition ( no- slip condition), are prescribed in at a solid wall both in the normal direction, and in particular in the tangential direction as the velocity is zero. This leads to the formation of a boundary layer, which is responsible for essential, molded only by the Navier -Stokes equations, phenomena. Only when the mean free path of moving molecules is large to the characteristic length of the geometry (eg for gases with extremely low densities or currents in extremely narrow gaps ), this condition is not meaningful.

In addition, must be prescribed on the edge yet either temperature or heat flux.

Solutions

Theoretical solution

It is still not succeeded in proving the existence of global solutions. Mathematicians like P.-L. Lions (see bibliography) look essentially the important special case of the incompressible Navier -Stokes equations. While here for the two-dimensional case, among others, by Roger Temam and Ciprian Foias already widespread existence, uniqueness and regularity results proved to be important, there is as yet no results for the general three-dimensional case, since some fundamental embedding theorems for so-called Sobolev spaces no longer be used can. However, there are for finite periods or special, especially small initial data in three-dimensional case - especially for weak solutions - existence and uniqueness results.

The problem of the general incompressible existence proof in three dimensions is, according to Clay Mathematics Institute of the most important unsolved mathematical problems at the time of the millennium.

In practice, one gets analytical solutions by the physical models / boundary conditions are simplified (special cases). Particular difficulty is here, the nonlinearity of the convective acceleration. Useful here is the presentation using the vorticity:

Closed analytic solutions exist for almost all cases in which the second term is zero. This is with the assumption that in three -dimensional flows, the vortex train is always along the power line (ie, the Helmholtz vortex theorem ), ie the case. This assumption is not true for all real flows.

The Navier -Stokes equations are ( the theory deals with existence and uniqueness of solutions, in most cases however, there is no closed-form solution formulas ) is an important application field of numerical mathematics. The portion that deals with the construction of numerical approximation methods for the Navier -Stokes equations, the computational fluid dynamics or computational fluid dynamics (CFD ).

Numerical Solution

In the numerical solution of the Navier -Stokes equations method of computational fluid mechanics are used. Be used for both finite difference, finite element and finite volume methods as well as for special tasks and spectral methods and other techniques as discretizations. The grid must in order to resolve the boundary layer correctly, can be extremely fine resolution in the normal direction near the wall. In the tangential direction is waived, so that the cells have extremely large aspect ratios on the wall.

The fine resolution enforces extremely small time steps for compliance with the CFL condition for explicit time integration. Therefore, implicit methods are generally used. Because of the nonlinearity of the equation system, the system must be solved iteratively (for example, multi-grid or Newton's method ). The combination of pulse and the continuity equation for the incompressible equations has a saddle-point structure may be utilized here.

A simple model for the simulation of liquids satisfies the Navier -Stokes equation in the hydrodynamic limit is FHP model. Its development leads to the Lattice -Boltzmann methods, which are particularly attractive in the context of parallelization to run on supercomputers.

In the field of computer graphics, several numerical solution methods have been used where by certain assumptions, a real-time display can be achieved, however partially, the physical correctness is not always granted. One example is developed by Jos Stam, " Stable fluid " process. Here, the Chorin'sche projection method for the field of computer graphics has been used.

Calculation of turbulent flows

To calculate turbulent flows, the Navier -Stokes equations can be directly calculated numerically. However, the resolution of the individual turbulence enforces a very fine grid, so this only in research with the help of supercomputers and at low Reynolds numbers is actually possible.

In practice, the solution of the Reynolds equations has prevailed. Here, however, a turbulence model is needed to close the system of equations.

As a middle course applies the Large Eddy Simulation that calculates at least the large eddy and direct numerical only simulates the small scales through a turbulence model.

Simplifications

Due to the difficult Lösbarkeitseigenschaften the Navier -Stokes equations, is in the applications ( as far as is physically meaningful ) try to look at simplified versions of the Navier -Stokes equations.

Euler equations

If the second order, such as friction, neglected terms ( η = 0; λ = 0), we obtain the Euler equations ( here for the incompressible case )

The compressible Euler equations in particular, play a role in aerodynamics as an approximation to the full Navier -Stokes equations.

Stokes equation

Another type of simplification is common in geodynamics, where the mantle of the earth (or other terrestrial planets) is treated as an extremely viscous liquid ( Creeping flow ). In this approximation is the diffusivity of the pulse, i.e., the kinematic viscosity, many orders of magnitude higher than the thermal diffusivity, and the inertia term can be neglected. This results in the Stokes equation:

Applying the projection to the Helmholtz equation, the pressure in the equation disappears:

With. This has the advantage that the equation depends only on. The original equation is obtained with

Is also referred to as Stokes operator.

On the other hand have a complicated geomaterials rheology, which leads to that the viscosity is not considered to be constant. For the incompressible case, this gives:

Boussinesq approximation

For gravity- dependent flows with small density variations and not to large temperature variations, the Boussinesq approximation is often used.