Classifications
Compressible vs. incompressible flow
All fluids are compressible to some extent; That is, changes in pressure or temperature cause changes in density. However, in many situations the pressure and temperature changes are small enough that the density changes are negligible. In this case, the flow can be modeled as an incompressible flow. Otherwise, the more general equations for compressible flow must be used.
Mathematically, incompressibility is expressed by saying that the density of a fluid parcel does not change as it moves in the flow field, i.e.
where is the material derivative, which is the sum of the local and the convective derivative. This additional constraint simplifies the governing equations, especially in the case where the fluid has a uniform density.
In the case of gas flow, to determine whether compressible or incompressible fluid dynamics is used, the Mach number of the flow is evaluated. As a rough guide, compressible effects can be ignored at Mach numbers below about 0.3. For liquids, the validity of the incompressibility assumption depends on the fluid properties (specifically, the critical pressure and temperature of the fluid) and the flow conditions (how close the actual flow pressure is to the critical pressure). Acoustics problems always require allowing for compressibility, since sound waves are compression waves that involve changes in the pressure and density of the medium through which they propagate.
Newtonian vs. non-Newtonian fluids
All fluids are viscous, meaning they exert some resistance to deformation: neighboring fluid elements moving at different speeds exert viscous forces on each other. The velocity gradient is called the strain rate; It has dimensions T. Isaac Newton showed that for many known fluids, such as water and air, the stress due to these viscous forces is linearly related to the strain rate. These fluids are called Newtonian fluids. The proportionality coefficient is called the viscosity of the fluid; for Newtonian fluids, it is a property of the fluid that is independent of strain rate.
Non-Newtonian fluids have more complicated and nonlinear stress-strain behavior. The subdiscipline of rheology describes the stress-strain behaviors of such fluids, including emulsions and slurries, some viscoelastic materials such as blood and some polymers, and sticky liquids such as latex, honey, and lubricants.[3].
Viscous flow versus Stokes flow
The dynamics of differential fluid elements is described with the help of Newton's second law. An accelerating parcel of fluid is subject to inertial effects.
The Reynolds number is a dimensionless quantity that characterizes the magnitude of inertial effects compared to the magnitude of viscous effects. A low Reynolds number () indicates that the viscous forces are very strong compared to the inertial forces. In such cases, inertial forces are sometimes neglected; This flow regime is called Stokes flow.
In contrast, high Reynolds numbers () indicate that inertial effects have more effect on the velocity field than viscous (friction) effects. In high Reynolds number flows, the flow is often modeled as an inviscid flow, an approximation in which viscosity is completely neglected. The elimination of viscosity allows the Navier-Stokes equations to be simplified into the Euler "Euler equations (fluids)"). The integration of Euler's equations along a streamline in an inviscid flow gives rise to the Bernoulli equation. When, in addition to being inviscid, the flow is irrotational everywhere, the Bernoulli equation can completely describe the flow everywhere. Such flows are called potential flow, because the velocity field can be expressed as the gradient of a potential energy expression.
This idea can work quite well when the Reynolds number is high. However, problems such as those involving solid limits may require viscosity to be included. Viscosity cannot be neglected near solid boundaries because the no-slip condition between the fluid and the solid boundary generates a thin region of high strain rate, the boundary layer, in which the effects of viscosity dominate and which, therefore, generates vorticity. Therefore, to calculate net forces on bodies (such as wings), viscous flow equations must be used: inviscid flow theory fails to predict drag forces "Drag (physics)"), a limitation known as D'Alembert's paradox.
One of the most widely used[4] model, especially in computational fluid dynamics, is to use two flow models: the Euler equations far from the body, and the boundary layer equations in a region close to the body. The two solutions can then be paired with each other, using the method of paired asymptotic expansions.
Steady versus non-steady flow
A flow that is not a function of time is called steady flow'. Steady-state flow refers to the condition in which the properties of the fluid at a point in the system do not change with time. Time-dependent flow is known as unstable (also called transient[6]). Whether a particular flow is stable or unstable may depend on the chosen frame of reference. For example, laminar flow over a sphere is stationary in the reference frame that is stationary with respect to the sphere. In a stationary reference frame with respect to a background flow, the flow is unstable.
Turbulent flows are unstable by definition. However, a turbulent flow can be statically stationary. The random velocity field is statistically stationary if all statistics are invariant under a shift in time.[7] This means, roughly, that all statistical properties are constant in time. Often the mean field is the object of interest, and this is constant also in a statistically steady flow.
Steady flows are usually more manageable than unsteady flows. The governing equations of a stationary problem have one dimension less (time) than the governing equations of the same problem without taking advantage of the stability of the flow field.
Laminar versus turbulent flow
Turbulence is a flow characterized by recirculation, vortices, and apparent randomness. Flow that does not present turbulence is called laminar. The presence of eddies or recirculation alone does not necessarily indicate turbulent flow; These phenomena may also be present in laminar flow. Mathematically, turbulent flow is usually represented by a Reynolds decomposition, in which the flow is decomposed into the sum of a mean component and a disturbance component.
It is believed that turbulent flows can be well described by using the Navier-Stokes equations. Direct numerical simulation (DNS), based on the Navier-Stokes equations, allows the simulation of turbulent flows at moderate Reynolds numbers. The restrictions depend on the power of the computer used and the effectiveness of the solution algorithm. DNS results have been found to agree well with experimental data for some flows[8].
Most flows of interest have Reynolds numbers too high for DNS to be a viable option,[7] given the state of computing power for the coming decades. Any flight vehicle large enough to carry a human (> 3 m), moving faster than 20 meters per second (72.0 km/h; 44.7 mph) is well above the DNS simulation limit (= 4 million). The wings of transport aircraft (such as on an Airbus A300 or a Boeing 747) have Reynolds numbers of 40 million (based on the chord dimension of the wing). Solving these real-life flow problems requires turbulence models for the foreseeable future. The Reynolds-averaged Navier-Stokes (RANS) equations combined with turbulence modeling provide a model of the effects of turbulent flow. This modeling primarily provides additional momentum transfer by Reynolds stresses, although turbulence also enhances heat and mass transfer. Another promising methodology is large eddy simulation (LES), especially in the form of separated eddy simulation (DES), which is a combination of RANS turbulence modeling and large eddy simulation.
Other approaches
There are a large number of other possible approaches to fluid dynamics problems. Some of the most used are listed below.
• - The Boussinesq approximation "Boussinesq approximation (buoyancy)")' neglects density variations except to calculate buoyancy forces. It is often used in free convection problems where density changes are small.
• - Lubrication Theory") and Hele-Shaw Flow")' exploit the large aspect ratio of the domain to show that certain terms in the equations are small and therefore can be neglected.
• - Thin body theory") is a methodology used in Stokes flow problems to estimate the force or flow field around a long, thin object in a viscous fluid.
• - Shallow water equations can be used to describe a layer of relatively inviscid fluid with a free surface, in which the surface gradient is small.
• - "Darcy's law" is used for flow in "porous media", and works with variables averaged over several pore widths.
• - In rotating systems, the quasi-geostrophic equations assume an almost perfect balance between pressure gradients and the Coriolis force. It is useful in the study of atmospheric dynamics.