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Navier-Stokes Equation

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Encyclopedia of Nonlinear Science

NAVIER-STOKES EQUATION

In 1822, the French engineer Claude Navier derived the Navier-Stokes equation, as an extension of Euler’s equation to include viscosity. Navier was initially interested in blood flow, and he used a molecular approach to arrive at the viscous terms. Navier’s equations were generalized to a compressible fluid by Poisson (1829), and one can find fully continuous derivations by De Saint-Venant (1843). A comprehensive treatment was given by George Stokes in 1845, who independently arrived at the results of Poisson and Saint-Venant using a continuous model. Stokes used the common assumption of linear relations between stress and strain rate and discovered “Stokes’ law” for the terminal velocity of objects descending in fluids, which he deduced from experiments with slowing pendulums in viscous media.

The Navier-Stokes equation derives from a general equation for the conservation of momentum, balancing forces per unit volume on both sides. It is given by

(1)

where υ(r, t) is the velocity vector in a cartesian coordinate system, ρ(r, t) is the density, g is the acceleration due to external forces (for example, gravitational, magnetic, electro-static forces), p(r, t) the pressure, and µ the viscosity coefficient. The material derivative is given by

(2)

where the first and second terms are the local time derivative of the quantity υ(r, t) with spatial coordinates fixed, and the convective (or advective) term accounting for the change of the quantity υ(r, t) at r due to the observer following the motion of the fluid with velocity υ. Both terms together are called the material derivative or convective derivative and are usually denoted by d/dt.

In an inertial coordinate system, the acceleration g originates solely from the gravitational potential of the Earth. Assuming a homogeneous mass at the center of the Earth, the acceleration g in Equation (1) is given by g=gk, where k is the local vertical direction, pointing to the center of gravity of the earth and g is the gravitational acceleration (982.1cm s⁻²), which is assumed constant in the first approximation. In a rotating coordinate system, the acceleration g consists of the centrifugal force and Coriolis force in addition to the gravitational acceleration, see Lamb (1906). Also, for large-scale processes, it is often necessary to account for tidal forces of the moon and sun. The term is the force per unit volume due to a pressure gradient of the scalar pressure field p(r, t) acting on an infinitesimal volume of fluid with infinitesimal mass.

The last two terms of Equation (1)

(3)

are the frictional terms which derive from a shear stress tensor for a Newtonian viscous fluid with constant viscosity coefficients. A fluid is called Newtonian if there is a linear relation between stress and rate of strain assuming isotropy, which is true for the most common conditions. Non-Newtonian fluids have more complex molecular structure or are mixtures of fluids.

The frictional term contains effects due to compression and rotation. If the fluid is incompressible then the divergence terms are missing from Equations (1) and (3), since υ=0. Often the coefficient of kinematic viscosity ν=µ/ρ is used instead of the viscosity µ. The viscosity of the fluid depends on temperature, and a table for typical values of viscosity µ, kinematic viscosity ν, and density ρ is shown in Table 1 for typical temperatures. If the temperature across the medium is not uniform then a variable viscosity may have to be accounted for (see Batchelor, 1970).

Table 1. Density ρ, viscosity µ and kinematic viscosity ν of air and water.

T(°C)	Air ρ(g cm⁻³)	μ(g cm⁻¹ s⁻¹)	ν(cm² s⁻¹)	Water ρ(g cm⁻³)	µ(g cm⁻¹ s⁻¹)	ν(cm² s⁻¹)
0	1.293×10⁻³	1.71×10⁻⁴	0.132	0.999	1.787×10⁻²	1.787×10⁻²
10	1.247×10⁻³	1.76×10⁻⁴	0.141	0.999	1.304×10⁻²	1.304×10⁻²
20	1.205×10⁻³	1.81×10⁻⁴	0.150	0.998	1.002×10⁻²	1.004×10⁻²

For an ideal (perfect) fluid without internal shear stress, the momentum equation reduces to Euler’s equation of motion for µ=0. Note that if the flow is irrotational and incompressible it essentially behaves as if it is inviscid (µ=0), because of the incompressibility condition (Note that a fluid may be incompressible yet ρ may not be constant.)

The system of Equations (1) comprises three momentum equations, together with an additional equation for mass conservation; a thermodynamic equation of state; a relation of pressure, density, and either temperature or entropy; and an energy equation for the additional thermodynamic variable (temperature or entropy). These provide six scalar equations for the determination of the six independent variables, velocity υ, density ρ, pressure p, and temperature or entropy as functions of space x and time t. If the fluid is incompressible, the equation of state becomes obsolete and only four equations are needed, the three momentum equations and the equation for conservation of mass.

Numerical Problems

As with most complicated nonlinear partial differential equations, the Navier-Stokes equation is solved numerically to model a specific fluid flow that is of interest experimentally and theoretically. Because there are many different ways to write the Navier-Stokes equation, the first numerical difficulty is to pick the formulation most suitable to the numerical technique that one wishes to employ and the dimensions of the model. One distinguishes between two-dimensional and three-dimensional models. For two-dimensional incompressible Navier-Stokes equation, for example, there are four different kinds of formulations: the primitive-variable (velocity and pressure), stream function-vorticity, stream function, and velocity-vorticity formulation.

It is important to distinguish between the real dissipation (µ≠0), given by the last term of Equation (1) and numerical dissipation, which is introduced by an accumulative error due to the limited order of accuracy of any numerical model. As the numerical dissipation of a good model is often negligible, it is sometimes appropriate to introduce an artificial viscosity that acts to damp growing high wave number modes, which can lead to numerical instability.

Euler’s equation conserves linear and angular momentum and energy, so it is necessary to verify that the numerical model conserves these conserved quantities and preserves symmetry. In cases where it is not possible to satisfy all these requirements, properties that are most essential to the physical problem are given priority. In order to satisfy such conserved quantities, it is important that the numerical scheme treats the nonlinear convective term

(4)

in a conservative manner, where represents the scalar quantity that is convected and could be one of the three velocity components or density ρ in the conservation of mass. Expression (4) is known as the convection form and is nonconservative numerically, since fluxes across mesh boundaries do not cancel. The convection form thus represents a major flaw of every numerical scheme. The most commonly used corresponding conservative formulation is

(5)

which is also known as the divergence form. Recently it was shown that the skewsymmetric form

has advantages, because it reduces aliasing errors (see below) for yet unknown reasons and is preferable to the popular rotational form

It is important to note that although both equations (4) and (5) are formally equivalent, they are not in a corresponding numerical scheme and care must be taken to avoid this common pitfall when modeling equations that contain convective terms such as the Navier-Stokes equation or the equation for mass conservation (see Hirsch, 2000).

Nonlinear terms such as also generate aliasing errors. These are high-frequency wave number modes appearing or being “aliased” as low-frequency modes that deteriorate the wave number spectrum and eventually lead to numerical blow-up.

Because a numerical model of the Navier-Stokes equation is coupled spatially and temporally, it is helpful to approximate the spatiotemporal scales of the dynamics. One should determine whether the spatial and temporal scales are equal to, smaller, or larger than each other. If, for example, the temporal scale is larger than the spatial scale, it is sufficient to use low order integration in time such as finite-difference methods, and if smaller or equal, then higher-order temporal integration such as spectral methods becomes necessary. In a temporally extensive computation, it is sometimes possible to revert to finite differences, choosing a relatively small time-step.

Especially difficult numerical problems arise when it is necessary to invert nondiagonal, nonsparse, nontrivial matrices, for example, in implicit time integrations or inverting the Laplacian. In such cases iterative techniques using preconditioning and multi-grid methods have been successfully employed. Complications arising from specifying the correct boundary conditions to use along boundaries of the physical domain should also be mentioned.

Although the Navier-Stokes equation is a very good model for a real fluid, it is important to have an understanding of the process or phenomenon that one wishes to model as well as a sound knowledge of the applicability of the numerical methods that one wishes to employ to be able to faithfully represent the physical dynamics of the fluid.

Phenomena

A salient feature of the Navier-Stokes equation is the phenomenon of turbulence. Due to viscous effects the flow of a real fluid can be observed in two very different states: a laminar state and a turbulent state.

Turbulence induces a cascade to fine scales that are eventually dissipated. It is important in many engineering applications, as a loss of energy means increased costs and can possibly cause damage to the structural body that comes in contact with the fluid or which moves through the fluid (e.g., optimization of airfoils, loss of lift force due to turbulence and boundary layer separation).

Closely related to the phenomenon of turbulence is boundary layer theory and boundary layer separation. Fluid flowing past a surface at rest relative to the fluid will experience friction at the surface of the material, and for the velocity field to be continuous, it is required that the normal as well as tangential fluid velocity vanish at the material surface (υ=0). For an idealized fluid (µ=0), only the normal velocity has to vanish.

It was the problem of matching the vanishing flow at the surface boundary to some nonzero fluid flow away from the boundary that led Prandtl (1905) to introduce the concept of boundary layer, which is a small “inner” fluid layer close to the surface where viscous effects dominate the flow. This inner layer is joined to an “outer” layer, where the fluid can be considered inviscid. The concept of boundary layer has led to great progress in fluid mechanics and perturbation methods, where matched asymptotic expansions feature prominently, but a complete understanding of boundary layer separation and its connection to turbulence is still missing.

A useful concept is vorticity, which can be regarded as a beautiful generalization of fluid motion. Formally vorticity is defined as

which simplifies to twice the angular velocity for solid body rotation. A fluid is called irrotational if its vorticity is zero and rotational if it is not. Taking the curl of Euler’s equation (µ=0) for an incompressible fluid in the absence of baroclinicity (and after some vector algebra) leads to the vorticity equation

Vortex lines and vortex strength are conserved, vortex lines move with the fluid, and an initially irrotational flow remains irrotational. Associated to vortices and turbulence is the, still not understood, phenomenon of vortex breakdown (Benjamin, 1978).

Among wave phenomena, the most striking is definitely the solitary wave, which appears as a nonlinear wave in a fluid internally as well as at fluid surfaces or interfaces. For free-surface wave phenomena, see Johnson (1997) and the references therein, and for a colorful overview, see Lighthill (1978). A beautiful picture gallery of fluid phenomena can be found in Van Dyke (1982). Other phenomena that can only be listed here include cavitation and thermal convection (Rayleigh-Bénard convection), sub- and supersonic flow, shock waves as well as most instability mechanisms and phenomena (Rayleigh-Taylor instability, Kelvin-Helmholtz instability, baro-tropic and baroclinic instability).

ANDREAS A.AIGNER

Navier-Stokes Equation

NAVIER-STOKES EQUATION

Numerical Problems

Phenomena

Further Reading