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Incompressible flow

Incompressible flow

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In fluid mechanics
Fluid mechanics
Fluid mechanics is the study of fluids and the forces on them. Fluid mechanics can be divided into fluid statics, the study of fluids at rest; fluid kinematics, the study of fluids in motion; and fluid dynamics, the study of the effect of forces on fluid motion...

 or more generally continuum mechanics
Continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modelled as a continuous mass rather than as discrete particles...

, incompressible (isochoric) flow refers to flow in which the material density is constant within an infinitesimal volume that moves with the velocity of the fluid. An equivalent statement that implies incompressible flow is that the divergence
Divergence
In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...

 of the fluid velocity is zero (see the derivation below, which illustrates why these conditions are equivalent).

Incompressible flow does not imply that the fluid itself is incompressible. It is shown in the derivation below that even compressible fluids can undergo incompressible flow. Incompressible fluids must have a constant density everywhere, while incompressible flow only requires that the density remain constant within a parcel of fluid which moves with the fluid velocity.

Derivation


The fundamental requirement for incompressible flow is that the density, , is constant within an infinitesimal volume, dV, which moves at the velocity of the fluid, v. Mathematically, this constraint implies that the material derivative (discussed below) of the density must vanish to ensure incompressible flow. Before introducing this constraint, we must apply the conservation of mass
Conservation of mass
The law of conservation of mass, also known as the principle of mass/matter conservation, states that the mass of an isolated system will remain constant over time...

 to generate the necessary relations. The mass is calculated by a volume integral of the density, :


The conservation of mass
Conservation of mass
The law of conservation of mass, also known as the principle of mass/matter conservation, states that the mass of an isolated system will remain constant over time...

 requires that the time derivative of the mass inside a control volume be equal to the mass flux, J, across its boundaries. Mathematically, we can represent this constraint in terms of a surface integral:


The negative sign in the above expression ensures that outward flow results in a decrease in the mass with respect to time, using the convention that the surface area vector points outward. Now, using the divergence theorem
Divergence theorem
In vector calculus, the divergence theorem, also known as Gauss' theorem , Ostrogradsky's theorem , or Gauss–Ostrogradsky theorem is a result that relates the flow of a vector field through a surface to the behavior of the vector field inside the surface.More precisely, the divergence theorem...

 we can derive the relationship between the flux and the partial time derivative of the density:


therefore:


The partial derivative of the density with respect to time need not vanish to ensure incompressible flow. When we speak of the partial derivative of the density with respect to time, we are referring to this rate of change within a control volume of fixed position. By allowing the partial time derivative of the density to be non-zero, we are not restricting ourselves to incompressible fluids because the density is allowed to change as observed from a fixed position as fluid flows through the control volume. This approach maintains generality, and not requiring that the partial time derivative of the density vanishes illustrates that compressible fluids can still undergo incompressible flow. What we are interested in now is the change in density of a control volume which moves along with the fluid velocity, v. The flux is related to the fluid velocity through the following function:


So that the conservation of mass implies that:


The previous relation (where we have used the appropriate product rule) is known as the continuity equation. Now, we need the following relation about the total derivative
Total derivative
In the mathematical field of differential calculus, the term total derivative has a number of closely related meanings.The total derivative of a function f, of several variables, e.g., t, x, y, etc., with respect to one of its input variables, e.g., t, is different from the partial derivative...

 of the density (where we apply the chain rule
Chain rule
In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...

):


So if we choose a control volume that is moving at the same rate as the fluid (i.e. (dx/dtdy/dtdz/dt) = v), then this expression simplifies to the material derivative:


And so using the continuity equation derived above, we see that:


A change in the density over time would imply that the fluid had either compressed or expanded (or that the mass contained in our constant volume, dV, had changed), which we have prohibited. We must then require that the material derivative of the density vanishes, and equivalently (for non-zero density) so must the divergence of the fluid velocity:


And so beginning with the conservation of mass
Conservation of mass
The law of conservation of mass, also known as the principle of mass/matter conservation, states that the mass of an isolated system will remain constant over time...

 and the constraint that the density within a moving volume of fluid remains constant, it has been shown that an equivalent condition required for incompressible flow is that the divergence of the fluid velocity vanishes.

Relation to compressibility


In some fields, a measure of the incompressibility of a flow is the change in density as a result of the pressure variations. This is best expressed in terms of the compressibility


If the compressibility is acceptably small, the flow is considered to be incompressible.

Relation to solenoidal field


An incompressible flow is described by a velocity field which is solenoidal. But a solenoidal field, besides having a zero divergence
Divergence
In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...

, also has the additional connotation of having non-zero curl (i.e., rotational component).

Otherwise, if an incompressible flow also has a curl of zero, so that it is also irrotational, then the velocity field is actually Laplacian.

Difference between incompressible flow and material


As defined earlier, an incompressible (isochoric) flow is the one in which
This is equivalent to saying that
i.e. the material derivative of the density is zero. Thus if we follow a material element, its mass density will remain constant. Note that the material derivative consists of two terms. The first term describes how the density of the material element changes with time. This term is also known as the unsteady term. The second term, describes the changes in the density as the material element moves from one point to another. This is the convection or the advection term. For a flow to be incompressible the sum of these terms should be zero.

On the other hand, a homogeneous, incompressible material is defined as one which has constant density throughout. For such a material, . This implies that, and independently.
From the continuity equation it follows that
Thus homogeneous materials always undergo flow that is incompressible, but the converse is not true.

It is common to find references where the author mentions incompressible flow and assumes that density is constant. Even though this is technically incorrect, it is an accepted practice. One of the advantages of using the incompressible material assumption over the incompressible flow assumption is in the momentum equation where the kinematic viscosity () can be assumed to be constant. The subtlety above is frequently a source of confusion. Therefore many people prefer to refer explicitly to incompressible materials or isochoric flow when being descriptive about the mechanics.

Related flow constraints


In fluid dynamics, a flow is considered to be incompressible if the divergence of the velocity is zero. However, related formulations can sometimes be used, depending on the flow system to be modelled. Some versions are described below:
  1. Incompressible flow: . This can assume either constant density (strict incompressible) or varying density flow. The varying density set accepts solutions involving small perturbations in density
    Density
    The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

    , pressure and/or temperature fields, and can allow for pressure stratification in the domain.
  2. Anelastic flow: . Principally used in the field of atmospheric sciences
    Atmospheric sciences
    Atmospheric sciences is an umbrella term for the study of the atmosphere, its processes, the effects other systems have on the atmosphere, and the effects of the atmosphere on these other systems. Meteorology includes atmospheric chemistry and atmospheric physics with a major focus on weather...

    , the anelastic constraint extends incompressible flow validity to stratified density and/or temperature as well as pressure. This allows the thermodynamic variables to relax to an 'atmospheric' base state seen in the lower atmosphere when used in the field of meteorology, for example. This condition can also be used for various astrophysical systems.
  3. Low Mach-number flow / Pseudo-incompressibility: . The low Mach-number
    Mach number
    Mach number is the speed of an object moving through air, or any other fluid substance, divided by the speed of sound as it is in that substance for its particular physical conditions, including those of temperature and pressure...

     constraint can be derived from the compressible Euler equations using scale analysis of non-dimensional quantities. The restraint, like the previous in this section, allows for the removal of acoustic waves, but also allows for large perturbations in density and/or temperature. The assumption is that the flow remains within a Mach number limit (normally less than 0.3) for any solution using such a constraint to be valid. Again, in accordance with all incompressible flows the pressure deviation must be small in comparison to the pressure base state.


These methods make differing assumptions about the flow, but all take into account the general form of the constraint for general flow dependent functions and .

Numerical approximations of incompressible flow


The stringent nature of the incompressible flow equations means that specific mathematical techniques have been devised to solve them. Some of these methods include:
  1. The projection method
    Projection method (fluid dynamics)
    The projection method is an effective means of numerically solving time-dependent incompressible fluid-flow problems. It was originally introduced by Alexandre Chorin in 1967 and independently by Roger Temam as an efficient means of solving the incompressible Navier-Stokes equations...

    (both approximate and exact)
  2. Artificial compressibility technique (approximate)
  3. Compressibility pre-conditioning