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Large-Eddy Simulation and Multi-Scale Methods. Fluids are composed of molecules that collide with one another and solid objects. However, the continuum assumption assumes that fluids are continuous, rather than discrete. The fact that the fluid is made up of discrete molecules is ignored. The equations can be simplified in a number of ways, all of which make them easier to solve. Some of the simplifications allow some simple fluid dynamics problems to be solved in closed form. A control volume is a discrete volume in space through which fluid is assumed to flow.
The integral formulations of the conservation laws are used to describe the change of mass, momentum, or energy within the control volume. The rate of change of fluid mass inside a control volume must be equal to the net rate of fluid flow into the volume. The left-hand side of the above expression is the rate of increase of mass within the volume and contains a triple integral over the control volume, whereas the right-hand side contains an integration over the surface of the control volume of mass convected into the system. Mass flow into the system is accounted as positive, and since the normal vector to the surface is opposite the sense of flow into the system the term is negated. In the above integral formulation of this equation, the term on the left is the net change of momentum within the volume.
The first term on the right is the net rate at which momentum is convected into the volume. The second term on the right is the force due to pressure on the volume’s surfaces. The following is the differential form of the momentum conservation equation. The equation above is a vector equation in a three-dimensional flow, but it can be expressed as three scalar equations in three coordinate directions. The viscous dissipation function governs the rate at which mechanical energy of the flow is converted to heat.
However, in many situations the changes in pressure and temperature are sufficiently small that the changes in density are negligible. This additional constraint simplifies the governing equations, especially in the case when the fluid has a uniform density. As a rough guide, compressible effects can be ignored at Mach numbers below approximately 0. All fluids are viscous, meaning that they exert some resistance to deformation: neighbouring parcels of fluid moving at different velocities exert viscous forces on each other.
Newtonian fluids, it is a fluid property that is independent of the strain rate. An accelerating parcel of fluid is subject to inertial effects. Bernoulli’s equation can completely describe the flow everywhere. This idea can work fairly well when the Reynolds number is high.
However, problems such as those involving solid boundaries may require that the viscosity be included. Steady-state flow refers to the condition where the fluid properties at a point in the system do not change over time. Whether a particular flow is steady or unsteady, can depend on the chosen frame of reference. In a frame of reference that is stationary with respect to a background flow, the flow is unsteady. This roughly means that all statistical properties are constant in time.
Steady flows are often more tractable than otherwise similar unsteady flows. It should be noted, however, that the presence of eddies or recirculation alone does not necessarily indicate turbulent flow—these phenomena may be present in laminar flow as well. Stokes equations, makes it possible to simulate turbulent flows at moderate Reynolds numbers. Restrictions depend on the power of the computer used and the efficiency of the solution algorithm.