The fundamental basis of almost all CFD problems are the Navier–Stokes equations, which define any single-phase fluid flow.  In physics, the Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances. These equations arise from applying Newton’s second law to fluid motion, together with the assumption that the fluid stress is the sum of a diffusing viscous term (proportional to the gradient of velocity), plus a pressure term.

Newton’s laws of motion are three physical laws that form the basis for classical mechanics. They describe the relationship between the forces acting on a body and its motion due to those forces. They have been expressed in several different ways over nearly three centuries, and can be summarized as follows:

  1. First law: The velocity of a body remains constant unless the body is acted upon by an external force.
  2. Second law: The acceleration a of a body is parallel and directly proportional to the net force F and inversely proportional to the mass m, i.e., F = ma.
  3. Third law: The mutual forces of action and reaction between two bodies are equal, opposite and collinear.

The three laws of motion were first compiled by Sir Isaac Newton in his work Philosophiæ Naturalis Principia Mathematica, first published in 1687. Newton used them to explain and investigate the motion of many physical objects and systems.

The second law states that the net force on a particle is equal to the time rate of change of its linear momentum p in an inertial reference frame:

\mathbf{F} = \frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t} = \frac{\mathrm{d}(m\mathbf v)}{\mathrm{d}t},

where, since the law is valid only for constant-mass systems, the mass can be taken outside the differentiation operator by the constant factor rule in differentiation. Thus,

\mathbf{F} = m\,\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t} = m\mathbf{a},

where F is the net force applied, m is the mass of the body, and a is the body’s acceleration. Thus, the net force applied to a body produces a proportional acceleration. In other words, if a body is accelerating, then there is a force on it.

Any mass that is gained or lost by the system will cause a change in momentum that is not the result of an external force. A different equation is necessary for variable-mass systems (see below).

Consistent with the first law, the time derivative of the momentum is non-zero when the momentum changes direction, even if there is no change in its magnitude; such is the case with uniform circular motion. The relationship also implies the conservation of momentum: when the net force on the body is zero, the momentum of the body is constant. Any net force is equal to the rate of change of the momentum.

Newton’s second law requires modification if the effects of special relativity are to be taken into account, because at high speeds the approximation that momentum is the product of rest mass and velocity is not accurate.

velocity field

The Navier–Stokes equations dictate not position but rather velocity. A solution of the Navier–Stokes equations is called a velocity field or flow field, which is a description of the velocity of the fluid at a given point in space and time. Once the velocity field is solved for, other quantities of interest (such as flow rate or drag force) may be found. This is different from what one normally sees in classical mechanics, where solutions are typically trajectories of position of a particle or deflection of a continuum. Studying velocity instead of position makes more sense for a fluid; however for visualization purposes one can compute various trajectories.


The Navier–Stokes equations are nonlinear partial differential equations in almost every real situation[2][3]. In some cases, such as one-dimensional flow and Stokes flow (or creeping flow), the equations can be simplified to linear equations. The nonlinearity makes most problems difficult or impossible to solve and is the main contributor to the turbulence that the equations model.

The nonlinearity is due to convective acceleration, which is an acceleration associated with the change in velocity over position. Hence, any convective flow, whether turbulent or not, will involve nonlinearity. An example of convective but laminar (nonturbulent) flow would be the passage of a viscous fluid (for example, oil) through a small converging nozzle. Such flows, whether exactly solvable or not, can often be thoroughly studied and understood.

steady state

A system in a steady state has numerous properties that are unchanging in time. This implies that for any property p of the system, the partial derivative with respect to time is zero:

\frac{\partial p}{\partial t} = 0

The concept of steady state has relevance in many fields, in particular thermodynamics and economics. Steady state is a more general situation than dynamic equilibrium. If a system is in steady state, then the recently observed behavior of the system will continue into the future. In stochastic systems, the probabilities that various states will be repeated will remain constant.

In many systems, steady state is not achieved until some time has elapsed after the system is started or initiated. This initial situation is often identified as a transient state, start-up or warm-up period.

While a dynamic equilibrium occurs when two or more reversible processes occur at the same rate, and such a system can be said to be in steady state, a system that is in steady state may not necessarily be in a state of dynamic equilibrium, because some of the processes involved are not reversible.

For example: The flow of fluid through a tube, or electricity through a network, could be in a steady state because there is a constant flow of fluid, or electricity. Conversely, a tank which is being drained or filled with fluid would be an example of a system in transient state, because the volume of fluid contained in it changes with time.

incompressible flow of newtonian fluids

A simplification of the resulting flow equations is obtained when considering an incompressible flow of a Newtonian fluid  (A Newtonian fluid -named after Isaac Newton– is a fluid whose stress versus strain rate curve is linear and passes through the origin. The constant of proportionality is known as the viscosity).  In fluid mechanics or more generally continuum mechanics, incompressible flow (isochoric flow) refers to a flow in which the material density is constant within a fluid parcel – an infinitesimal volume that moves with the velocity of the fluid. An equivalent statement implying incompressibility is, that the divergence of the fluid velocity is zero.

An isochoric process, also called a constant-volume process, an isovolumetric process, or an isometric process, is a thermodynamic process during which the volume of the closed system undergoing such a process remains constant. An isochoric process is exemplified by the heating or the cooling of the contents of a sealed, inelastic container: The thermodynamic process is the addition or removal of heat; the isolation of the contents of the container establishes the closed system; and the inability of the container to deform imposes the constant-volume condition.

Incompressible flow does not imply that the fluid itself is incompressible. It is shown in the derivation below that (under the right conditions) even compressible fluids can – to good approximation – be modelled as an incompressible flow. Incompressible flow implies that the density remains constant within a parcel of fluid which moves with the fluid velocity.

The assumption of incompressibility rules out the possibility of sound or shock waves to occur; so this simplification is invalid if these phenomena are important. The incompressible flow assumption typically holds well even when dealing with a “compressible” fluid — such as air at room temperature — at low Mach numbers (even when flowing up to about Mach 0.3). Taking the incompressible flow assumption into account and assuming constant viscosity, the Navier–Stokes equations will read, in vector form:

Navier–Stokes equations (Incompressible flow)\rho \left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v}\right) = -\nabla p + \mu \nabla^2 \mathbf{v} + \mathbf{f}.

Here f represents “other” body forces (forces per unit volume), such as gravity or centrifugal force. The shear stress term \scriptstyle \nabla \boldsymbol{\mathsf{T}} becomes the useful quantity \scriptstyle \mu \nabla^2 \mathbf{v} (\scriptstyle \nabla^2 is the vector Laplacian) when the fluid is assumed incompressible, homogeneous and Newtonian, where \scriptstyle \mu is the (constant) dynamic viscosity.

It’s well worth observing the meaning of each term (compare to the Cauchy momentum equation):

<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />
\overbrace{\rho \Big(<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />
\underbrace{\frac{\partial \mathbf{v}}{\partial t}}_{<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />
\begin{smallmatrix}<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />
  \text{Unsteady}\\<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />
  \text{acceleration}<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />
\end{smallmatrix}} +<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />
\underbrace{\mathbf{v} \cdot \nabla \mathbf{v}}_{<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />
\begin{smallmatrix}<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />
  \text{Convective} \\<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />
  \text{acceleration}<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />
\end{smallmatrix}}\Big)}^{\text{Inertia (per volume)}} =<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />
\overbrace{\underbrace{-\nabla p}_{<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />
\begin{smallmatrix}<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />
  \text{Pressure} \\<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />
  \text{gradient}<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />
\end{smallmatrix}} +<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />
\underbrace{\mu \nabla^2 \mathbf{v}}_{\text{Viscosity}}}^{\text{Divergence of stress}} +<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />
\underbrace{\mathbf{f}}_{<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />
\begin{smallmatrix}<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />
  \text{Other} \\<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />
  \text{body} \\<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />
  \text{forces}<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />
\end{smallmatrix}}.<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />

Note that only the convective terms are nonlinear for incompressible Newtonian flow. The convective acceleration is an acceleration caused by a (possibly steady) change in velocity over position, for example the speeding up of fluid entering a converging nozzle. Though individual fluid particles are being accelerated and thus are under unsteady motion, the flow field (a velocity distribution) will not necessarily be time dependent.

Another important observation is that the viscosity is represented by the vector Laplacian of the velocity field (interpreted here as the difference between the velocity at a point and the mean velocity in a small volume around). This implies that – for a Newtonian fluid – viscosity operates in a diffusion of momentum, in much the same way as the diffusion of heat seen in the heat equation (which also involves the Laplacian).

If temperature effects are also neglected, the only “other” equation (apart from initial/boundary conditions) needed is the mass continuity equation. Under the assumption of incompressibility, the density of a fluid parcel is constant and it follows that the continuity equation will simplify to:

\nabla \cdot \mathbf{v} = 0.

This is more specifically a statement of the conservation of volume (see divergence and isochoric process).

These equations are commonly used in 3 coordinates systems: Cartesian, cylindrical, and spherical. While the Cartesian equations seem to follow directly from the vector equation above, the vector form of the Navier–Stokes equation involves some tensor calculus which means that writing it in other coordinate systems is not as simple as doing so for scalar equations (such as the heat equation).


Turbulence is the time dependent chaotic behavior seen in many fluid flows. It is generally believed that it is due to the inertia of the fluid as a whole: the culmination of time dependent and convective acceleration; hence flows where inertial effects are small tend to be laminar (the Reynolds number quantifies how much the flow is affected by inertia). It is believed, though not known with certainty, that the Navier–Stokes equations describe turbulence properly.

The numerical solution of the Navier–Stokes equations for turbulent flow is extremely difficult, and due to the significantly different mixing-length scales that are involved in turbulent flow, the stable solution of this requires such a fine mesh resolution that the computational time becomes significantly infeasible for calculation (see Direct numerical simulation). Attempts to solve turbulent flow using a laminar solver typically result in a time-unsteady solution, which fails to converge appropriately. To counter this, time-averaged equations such as the Reynolds-averaged Navier–Stokes equations (RANS), supplemented with turbulence models, are used in practical computational fluid dynamics (CFD) applications when modeling turbulent flows. Some models include the Spalart-Allmaras, k-ω (k-omega), k-ε (k-epsilon), and SST models which add a variety of additional equations to bring closure to the RANS equations. Another technique for solving numerically the Navier–Stokes equation is the Large eddy simulation (LES). This approach is computationally more expensive than the RANS method (in time and computer memory), but produces better results since the larger turbulent scales are explicitly resolved.

In fluid dynamics, turbulence kinetic energy (TKE) is the mean kinetic energy per unit mass associated with eddies in turbulent flow. Physically, the turbulence kinetic energy is characterised by measured root-mean-square (RMS) velocity fluctuations.

Generally, the TKE can be quantified by the mean of the turbulence normal stresses:

 k = \frac12 \left( \overline{(u'_1)^2} + \overline{(u'_2)^2} + \overline{(u'_3)^2} \right).

TKE can be produced by fluid shear, friction or buoyancy, or through external forcing at low-frequency eddie scales(integral scale). Turbulence kinetic energy is then transferred down the turbulence energy cascade, and is dissipated by viscous forces at the Kolmogorov scale. This process of production, transport and dissipation can be expressed as:

 \frac{Dk}{Dt} + \nabla \cdot T' = P - \epsilon,

where: [1]

  •  Dk/Dt is the mean-flow material derivative of TKE;
  •  \nabla \cdot T' is the turbulence transport of TKE;
  •  P is the production of TKE, and
  •  \epsilon is the TKE dissipation.

The full form of the TKE equation is

<br /><br /><br /><br /><br /><br /><br /><br />
\underbrace{ \frac{\partial k}{\partial t}}_{ \begin{smallmatrix}\text{Local}\\\text{derivative}\end{smallmatrix}}<br /><br /><br /><br /><br /><br /><br /><br />
+<br /><br /><br /><br /><br /><br /><br /><br />
\underbrace{\overline{u}_j \frac{\partial k}{\partial x_j}}_{ \begin{smallmatrix}\text{Advection}\end{smallmatrix}}<br /><br /><br /><br /><br /><br /><br /><br />
= -<br /><br /><br /><br /><br /><br /><br /><br />
\underbrace{ \frac{1}{\rho_o} \frac{\partial \overline{u'_i p'}}{\partial x_i} 	} _{ \begin{smallmatrix}\text{Pressure}\\\text{diffusion}\end{smallmatrix}}<br /><br /><br /><br /><br /><br /><br /><br />
-<br /><br /><br /><br /><br /><br /><br /><br />
\underbrace{ \frac{\partial \overline{k u_i}}{\partial x_j} 	}_{ \begin{smallmatrix}										\text{Turbulent}\\											\text{transport} \\											\mathcal{T}											\end{smallmatrix}}<br /><br /><br /><br /><br /><br /><br /><br />
	+ \underbrace{ \nu\frac{\partial^2 k}{\partial x^2_j} 						}_{\begin{smallmatrix}										\text{Molecular}\\										\text{viscous}\\										\text{transport}											\end{smallmatrix}}<br /><br /><br /><br /><br /><br /><br /><br />
	\underbrace{ - \overline{u'_i u'_j}\frac{\partial \overline{u_i}}{\partial x_j} 		}_{\begin{smallmatrix}										\text{Production}\\											\mathcal{P}												\end{smallmatrix}}<br /><br /><br /><br /><br /><br /><br /><br />
	- \underbrace{ \nu \overline{\frac{\partial u'_i}{\partial  x_j}\frac{\partial u'_i}{\partial x_j}} 											}_{\begin{smallmatrix}												\text{Dissipation}\\													\epsilon_k													\end{smallmatrix}}<br /><br /><br /><br /><br /><br /><br /><br />
	- \underbrace{ \frac{g}{\rho_o} \overline{\rho' u'_i}\delta_{i3}				}_{\begin{smallmatrix}													\text{Buoyancy flux}\\													b													\end{smallmatrix}}<br /><br /><br /><br /><br /><br /><br /><br />

By examining these phenomena, the turbulence kinetic energy budget for a particular flow can be found

In computational fluid dynamics (CFD), it is impossible to numerically simulate turbulence without discretising the flow-field as far as the Kolmogorov microscales, which is called direct numerical simulation (DNS). Because DNS simulations are exorbitantly expensive due to memory, computational and storage overheads, turbulence models are used to simulate the effects of turbulence. A variety of models are used, but generally TKE is a fundamental flow property which must be calculated in order for fluid turbulence to be modelled.

Reynolds-averaged Navier–Stokes (RANS) simulations use the Boussinesq eddy viscosity hypothesisto calculate the Reynolds stresses that result from the averaging procedure:

 \overline{u'_i u'_j} = 2/3 k \delta_{ij} - \nu_t \left( \frac{\partial \overline{u_i}}{\partial x_j} + \frac{\partial \overline{u_j}}{\partial x_i} \right),


 \nu_t = c \cdot k^{1/2}  l_m.

The exact method of resolving TKE depends upon the turbulence model used; k-ε (k–epsilon) models assume isotropy of turbulence whereby the normal stresses are equal:

 \overline{u'^2} = \overline{v'^2} = \overline{w'^2}.

This assumption makes modelling of turbulence quantities (k and \epsilon) simpler, but will not be accurate in scenarios where anisotropic behaviour of turbulence stresses dominates, and the implications of this in the production of turbulence also leads to over-prediction since the production depends on the mean rate of strain, and not the difference between the normal stresses (as they are, by assumption, equal) .

shear velocity

Shear velocity, also called friction velocity, is a form by which a shear stress may be re-written in units of velocity. It is useful as a method in fluid mechanics to compare true velocities, such as the velocity of a flow in a stream, to a velocity that relates shear between layers of flow.

Shear velocity is used to describe shear-related motion in moving fluids. It is used to describe:

  • Diffusion and dispersion of particles, tracers, and contaminants in fluid flows
  • The velocity profile near the boundary of a flow (see Law of the wall)
  • Transport of sediment in a channel

Shear velocity also helps in thinking about the rate of shear and dispersion in a flow. Shear velocity scales well to rates of dispersion and bedload sediment transport. A general rule is that the shear velocity is about 1/10 of the mean flow velocity.


Where \tau is the shear stress in an arbitrary layer of fluid and \rho is the density of the fluid.

Typically, for sediment transport applications, the shear velocity is evaluated at the lower boundary of an open channel:


Where \tau_b is the shear stress given at the boundary.

Shear velocity can also be defined in terms of the local velocity and shear stress fields (as opposed to whole-channel values, as given above).

law of the wall

In fluid dynamics, the law of the wall states that the average velocity of a turbulent flow at a certain point is proportional to the logarithm of the distance from that point to the “wall”, or the boundary of the fluid region. This law of the wall was first published by Theodore von Kármán, in 1930. It is only technically applicable to parts of the flow that are close to the wall (<20% of the height of the flow), though it is a good approximation for the entire velocity profile of natural streams.


An isothermal process is a change of a system, in which the temperature remains constant: ΔT = 0. This typically occurs when a system is in contact with an outside thermal reservoir (heat bath), and the change occurs slowly enough to allow the system to continually adjust to the temperature of the reservoir through heat exchange.


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