Boussinesq assumption

Bourdon tube Boussinesq assumption Bovatec Bovine albumin Bovine brain... 

Later we shall include combustion and flame radiation effects, but we will still maintain all of assumptions 2 to 5 above. The top-hat profile and Boussinesq assumptions serve only to simplify our mathematics, while retaining the basic physics of the problem. However, since the theory can only be taken so far before experimental data must be relied on for its missing pieces, the degree of these simplifications should not reduce the generality of the results. We shall use the following conservation equations in control volume form for a fixed CV and for steady state conditions ... 

Attention will initially be restricted to two-dimensional steady laminar flow. The Boussinesq assumptions will again be used and, consistent with this assumption, dissipation effects will be neglected. The governing equations, expressed in Cartesian coordinates, are then [1] ... 

The basic idea of RANS models is to account for the change in the fluid transport properties by introducing an eddy viscosity, I, also called turbulence viscosity, which relates the Reynolds stress tensor R to the fluid deformation. Such a relationship was first proposed by Boussinesq in the nineteenth century. More formally, this Boussinesq assumption can be written as... 

The mass flow rate in terms of the Boussinesq top-hat assumption is given by... 

Surface-tension forces act in a direction tangent to the interface. Boussinesq (B6) assumed, for a surface undergoing dilatation, that there must also be another force acting normal to the interface. Utilizing the same assumptions as Hadamard, he arrived at a Stokes-law correction factor of... 

As we have stated, the Boussinesq equations are based on the assumption that the material dealt with is elastic and isotropic. Of course, this... 

In dealing with Boussinesq s problem, Lord Rayleigh used the amount of heat H (measuring unit calorie) as one of the then used base dimensions. Only since the introduction of SI (Systeme International d Unites) it was required to make no distinction between heat and mechanical energy, because both were considered to be equal. In order to comply with this requirement, the Joule equivalent of heat J [M L2 T2 H-1] had to be introduced as a natural constant in the relevance list. If we proceed from the assumption of an inviscid , ideal liquid, no mechanical heat can be converted into heat. In this case, J is irrelevant. 

It will be assumed here that the flow in the enclosure is steady and remains laminar. It will also be assumed that the flow is two-dimensional. With these assumptions and using the Boussinesq approximation that was discussed earlier, the equations governing the flow in the enclosure are ... 

The density variation due to the temperature generates a buoyancy force in the gravitational field. However, this has little influence on the other forces, including inertia and friction, which affect the fluid particles. As a good approximation it is sufficient to consider the temperature dependence of the density in buoyancy alone. This assumption is known as the Boussinesq-Approximation8. A characteristic (volume related) lift force is... 

The equations do not make any statement as to how the turbulent contributions are calculated. An additional assumption has to be made for this. A particularly simple rule was made by J.V. Boussinesq (1842-1929), who suggested the following for the Reynolds stress... 

As the Reynolds stress has to disappear approaching the wall, the turbulent viscosity cannot be constant. Flows adjacent to the wall cannot be described by Boussinesq s rule, when et = const is presumed. However for flows like those which occur in turbulent free jets, the assumption of constant turbulent viscosity is highly suitable. Corresponding to the Boussinesq rule a turbulent thermal diffusivitf at (SI units m2/s) is introduced, through... 

We will now deal with free flow on a vertical, flat wall whose temperature d0 is constant and larger than the temperature in the semi-infinite space. The coordinate origin lies, in accordance with Fig. 3.43, on the lower edge, the coordinate x runs along the wall, with y normal to it. Steady flow will be presumed. All material properties are constant. The density will only be assumed as temperature dependent in the buoyancy term, responsible for the free flow, in the momentum equation, in all other terms it is assumed to be constant. These assumptions from Oberbeck (1879) and Boussinesq (1903), [3.45], [3.46] are also known as the Boussinesq approximation although it would be more correct to speak of the Oberbeck-Boussinesq approximation. It takes into account that the locally variable density is a prerequisite for free flow. The momentum equation (3.294) in... 

Although numerous turbulence models are reported in the literature,1113 by far the most popular is the two-equation k-e model, first proposed by Jones and Launder.14 In this model, the turbulent stresses are recast in a form similar to the molecular stress tensor with mean velocity gradients, an assumption generally known as the Boussinesq hypothesis ... 

Circulation models are based on the equations of motion of the geophysical fluid dynamics and on the thermodynamics of seawater. The model area is divided into finite size grid cells. The state of the ocean is described by the velocity, temperature, and salinity in each grid cell, and its time evolution can be computed from the three-dimensional model equations. To reduce the computational demands, the model ocean is usually incompressible and the vertical acceleration is neglected, the latter assumption is known as hydrostatic approximation. This removes sound waves in the ocean from the model solution. In the horizontal equations, the Boussinesq approximation is applied and small density changes are ignored except in the horizontal pressure gradient terms. This implies that such models conserve... 

To simplify the problems represented by Eqs. (2 88) and (2 93), one of two possible assumptions is normally introduced. If the bounding surfaces are all at the same temperature, so that the only source of heat is viscous dissipation, it is often a good approximation to assume that the fluid is isothermal (i.e., the temperature is a constant, independent of spatial position or time). In this case, we do not need to consider (2 93) at all because it is an equation that is to be used to determine the temperature as a function of position and time. In addition, as noted in Chap. 2, the equation of motion also simplifies when the fluid is isothermal to either the form (2-89) or to (2 91) if the density in the system is also a constant, independent of position. When the temperature at the bounding surfaces is not constant, a different approximation known as the Boussinesq approximation is often used to simplify the problem. A detailed discussion of this approximation is postponed to a later point in the book. Here, we simply note the basic idea, which is that the material properties may be approximated as constants, provided that the temperature changes are not too large. In this case, the values of these properties can be evaluated at a representative temperature of the system, such as its mean value. An exception, as we shall see later, is that we sometimes cannot ignore the spatial variations of the density p in the body-force term of (2-88) even when the temperature changes are modest. Such density variations can produce motion... 

To do this, we assume that the temperature difference AT = TH — Tc is small enough that the Boussinesq approximation, first introduced in Chap. 2, can be employed. By this assumption, we neglect all of the temperature variation of material properties apart from the variation of the interfacial tension, as already indicated by Eq. (6 200). To be specific, we may assume that the other material parameters, such as /x, p, k, Cp, etc., are evaluated at... 

Of course, (12-164) is still exact, and the system of equations (12-164), (12-160), and (12-161) is no easier to solve than the original system of equations. To produce a tractable problem for analytic solution, it is necessary to introduce the so-called Boussinesq approximation, which has been used for many of the existing analyses of natural and mixed convection problems. The essence of this approximation is the assumption that the temperature variations in the fluid are small enough that the material properties p, p, k, and Cp can be approximated by their values at the ambient temperature Tq, except in the body-force term in (12-164), where the approximation p = po would mean that the fluid remains motionless. 

The Basset-Boussinesq-Oseen (BBO) equation can be simplified for gas-particle flows with a very small density ratio between the carrier phase and the discrete phase ( 10" ) and with the assumption of one-way coupling such that only the carrier phase has influence on the particle but not vice versa. The particles are governed by these nondimensional motion equations ... 

The theoretically simplest approach is to assume that no external boundary layer exists, thus obtaining the highest average values for the external transfer coefficients. Experience proves this assumption to hold for many important practical applications (D8, H8, HI3, W3, and others), where the Reynolds number and viscosity ratio Hcl ld are high. Boussinesq (BIO) was the first to... 

The tidal current fields at the project sea area are simulated by the three-dimensional numerical model (MIKE3 FM) which developed by Danish Hydraulics Research Institute. The model is based on the solution of three-dimensional incompressible Reynolds Navier-Stokes equations, subject to the assumption of Boussinesq and hydrostatic pressure. 

Boussinesq s theory (Boussinesq 1885) was developed at the end of the 19th century. Even though Boussinesq s theory is not currently used in multi-layer pavement engineering, it was the basis for the development of all subsequent stress/strain distribution theories. Boussinesq s theory refers to only one layer of uniform and homogeneous material such as, by assumption, soil material-subgrade. 

In Boussinesq s original concept, the eddy viscosity was assumed to be spatially constant. However, this assumption is valid only if the turbulent flow field is homogeneous, which is not frequent. For general cases, it is necessary to consider the variation of turbulent viscosity along the flow. 

In 1885 Joseph Boussinesq (6), trying to extend the validity of these results to the case of axi-symetrical rigid convex punches indenting a flat semi-infinite elastic medium, demonstrates that, without an adequate boundary comlition, the size of the contact area is generally unknown. In ordo to overcome this difficulty, he imposes that normal stresses vanish on the border of the contact area. In other words, the profile of the distorted medium must be tangent to the surfiice of the punch on the border of the contact area. Note that this condition is the same as the condition presupposed by the Hertz s theory. With this assumption, the size oh of the contact area and the penetration depth 5h are completely defined (Figure 1). 

Boussinesq Hypothesis. The Boussinesq hypothesis makes the assumption that the Reynolds stresses can be expressed in terms of mean velocity gradients. The following statement of the hypothesis shows the introduction of a new constant that is dimensionally equivalent to viscosity ... 

---