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While the Boussinesq approximation is applicable to fairly long waves – that is, when the wavelength is large compared to the water depth – the Stokes expansion is more appropriate for short waves (when the wavelength is of the same order as the water depth, or shorter).
The essential idea in the Boussinesq approximation is the elimination of the vertical coordinate from the flow equations, while retaining some of the influences of the vertical structure of the flow under water waves. This is useful because the waves propagate in the horizontal plane and have a different (not wave-like) behaviour in the vertical direction. Often, as in Boussinesq's case, the interest is primarily in the wave propagation.
This elimination of the vertical coordinate was first done by Joseph Boussinesq in 1871, to construct an approximate solution for the solitary wave (or wave of translation). Subsequently, in 1872, Boussinesq derived the equations known nowadays as the Boussinesq equations.
The steps in the Boussinesq approximation are:
Thereafter, the Boussinesq approximation is applied to the remaining flow equations, in order to eliminate the dependence on the vertical coordinate.
As a result, the resulting partial differential equations are in terms of functions of the horizontal coordinates (and time).
As an example, consider potential flow over a horizontal bed in the ("x,z") plane, with "x" the horizontal and "z" the vertical coordinate. The bed is located at , where "h" is the mean water depth. A Taylor expansion is made of the velocity potential "φ(x,z,t)" around the bed level :
where "φb(x,t)" is the velocity potential at the bed. Invoking Laplace's equation for "φ", as valid for incompressible flow, gives:
since the vertical velocity is zero at the – impermeable – horizontal bed . This series may subsequently be truncated to a finite number of terms.
For water waves on an incompressible fluid and irrotational flow in the ("x","z") plane, the boundary conditions at the free surface elevation are:
Now the Boussinesq approximation for the velocity potential "φ", as given above, is applied in these boundary conditions. Further, in the resulting equations only the linear and quadratic terms with respect to "η" and "ub" are retained (with the horizontal velocity at the bed ). The cubic and higher order terms are assumed to be negligible. Then, the following partial differential equations are obtained:
This set of equations has been derived for a flat horizontal bed, "i.e." the mean depth "h" is a constant independent of position "x". When the right-hand sides of the above equations are set to zero, they reduce to the shallow water equations.
Under some additional approximations, but at the same order of accuracy, the above set A can be reduced to a single partial differential equation for the free surface elevation "η":
From the terms between brackets, the importance of nonlinearity of the equation can be expressed in terms of the Ursell number.
In dimensionless quantities, using the water depth "h" and gravitational acceleration "g" for non-dimensionalization, this equation reads, after normalization:
Water waves of different wave lengths travel with different phase speeds, a phenomenon known as frequency dispersion. For the case of infinitesimal wave amplitude, the terminology is "linear frequency dispersion". The frequency dispersion characteristics of a Boussinesq-type of equation can be used to determine the range of wave lengths, for which it is a valid approximation.
The linear frequency dispersion characteristics for the above set A of equations are:
The relative error in the phase speed "c" for set A, as compared with linear theory for water waves, is less than 4% for a relative wave number . So, in engineering applications, set A is valid for wavelengths "λ" larger than 4 times the water depth "h".
The linear frequency dispersion characteristics of equation B are:
The relative error in the phase speed for equation B is less than 4% for , equivalent to wave lengths "λ" longer than 7 times the water depth "h", called fairly long waves.
For short waves with equation B become physically meaningless, because there are no longer real-valued solutions of the phase speed.
The original set of two partial differential equations (Boussinesq, 1872, equation 25, see set A above) does not have this shortcoming.
The shallow water equations have a relative error in the phase speed less than 4% for wave lengths "λ" in excess of 13 times the water depth "h".
There are an overwhelming number of mathematical models which are referred to as Boussinesq equations. This may easily lead to confusion, since often they are loosely referenced to as "the" Boussinesq equations, while in fact a variant thereof is considered. So it is more appropriate to call them Boussinesq-type equations. Strictly speaking, "the" Boussinesq equations is the above-mentioned set B, since it is used in the analysis in the remainder of his 1872 paper.
Some directions, into which the Boussinesq equations have been extended, are:
While the Boussinesq equations allow for waves traveling simultaneously in opposing directions, it is often advantageous to only consider waves traveling in one direction. Under small additional assumptions, the Boussinesq equations reduce to:
Besides solitary wave solutions, the Korteweg–de Vries equation also has periodic and exact solutions, called cnoidal waves. These are approximate solutions of the Boussinesq equation.
For the simulation of wave motion near coasts and harbours, numerical models – both commercial and academic – employing Boussinesq-type equations exist. Some commercial examples are the Boussinesq-type wave modules in MIKE 21 and SMS. Some of the free Boussinesq models are Celeris, COULWAVE, and FUNWAVE. Most numerical models employ finite-difference, finite-volume or finite element techniques for the discretization of the model equations. Scientific reviews and intercomparisons of several Boussinesq-type equations, their numerical approximation and performance are e.g. , and .
In fluid dynamics, Batchelor vortices, first described by George Batchelor in a 1964 article, have been found useful in analyses of airplane vortex wake hazard problems.
The Batchelor vortex is an approximate solution to the Navier-Stokes equations obtained using a boundary layer approximation. The physical reasoning behind this approximation is the assumption that the axial gradient of the flow field of interest is of much smaller magnitude than the radial gradient.
The axial, radial and azimuthal velocity components of the vortex are denoted formula_1,formula_2 and formula_3 respectively and can be represented in cylindrical coordinates formula_4 as follows:<br>
The parameters in the above equations are
Note that the radial component of the velocity is zero and that the axial and azimuthal components depend only on formula_13.
We now write the system above in dimensionless form by scaling time by a factor formula_14. Using the same symbols for the dimensionless variables, the Batchelor vortex can be expressed in terms of the dimensionless variables as
where formula_16 denotes the free stream axial velocity and formula_17 is the Reynolds number.
If one lets formula_18 and considers an infinitely large swirl number then the Batchelor vortex simplifies to the Lamb–Oseen vortex for the azimuthal velocity:
When using the notation formula_1 for dynamic viscosity, formula_2 for the liquid-solid contact angle, formula_3 for surface tension , formula_4 for the fluid density, "t" for time, and "r" for the cross-sectional radius of the capillary and "x" for the distance the fluid has advanced, the Bosanquet equation of motion is
assuming that the motion is completely driven by surface tension, with no applied pressure at either end of the capillary tube.
The solution of the Bosanquet equation can be split into two timescales, firstly to account for the initial motion of the fluid by considering a solution in the limit of time approaching 0 giving the form
For the condition of short time this shows a meniscus front position proportional to time rather than the Lucas-Washburn square root of time, and the independence of viscosity demonstrates plug flow.
As time increases after the initial time of acceleration, the equation decays to the familiar Lucas-Washburn form dependent on viscosity and the square root of time.
The black-oil equations are a set of partial differential equations that describe fluid flow in a petroleum reservoir, constituting the mathematical framework for a black-oil reservoir simulator.
The term "black-oil" refers to the fluid model, in which water is modeled explicitly together with two hydrocarbon components, one (pseudo) oil phase and one (pseudo-)gas phase.
This is in contrast with a compositional formulation, in which each hydrocarbon component (arbitrary number) is handled separately
The equations of an extended black-oil model are
formula_4 is a porosity of the porous medium,
and vapor ("gas") phases in the reservoir,
Darcy velocities of the liquid phase, water phase and vapor phase in the reservoir.
The oil and gas at the surface (standard conditions) could be produced from both liquid and vapor phases existing at high pressure and temperature of reservoir conditions. This is characterized by the following quantities:
(ratio of some volume of reservoir liquid
to the volume of oil at standard conditions
obtained from the same volume of reservoir liquid),
formula_9 is a water formation volume factor
(ratio of volume of water at reservoir conditions to volume of water at standard conditions),
formula_10 is a gas formation volume factor
(ratio of some volume of reservoir vapor
to the volume of gas at standard conditions obtained from the same volume of reservoir vapor),
formula_11 is a solution of gas in oil phase
(ratio of volume of gas to the volume of oil at standard conditions
obtained from some amount of liquid phase at reservoir conditions),
formula_12 is a vaporized oil in gas phase
(ratio of volume of oil to the volume of gas at standard conditions
obtained from some amount of vapor phase at reservoir conditions).
In fluid dynamics, the Oseen equations (or Oseen flow) describe the flow of a viscous and incompressible fluid at small Reynolds numbers, as formulated by Carl Wilhelm Oseen in 1910. Oseen flow is an improved description of these flows, as compared to Stokes flow, with the (partial) inclusion of convective acceleration.
Oseen's work is based on the experiments of G.G. Stokes, who had studied the falling of a sphere through a viscous fluid. He developed a correction term, which included inertial factors, for the flow velocity used in Stokes' calculations, to solve the problem known as Stokes' paradox. His approximation leads to an improvement to Stokes' calculations.
The Oseen equations are, in case of an object moving with a steady flow velocity U through the fluid—which is at rest far from the object—and in a frame of reference attached to the object:
The boundary conditions for the Oseen flow around a rigid object are:
with "r" the distance from the object's center, and "p"∞ the undisturbed pressure far from the object.
A fundamental property of Oseen's equation is that the general solution can be split into "longitudinal" and "transversal" waves.
A solution formula_3 is a longitudinal wave if the velocity is irrotational and hence the viscous term drops out. The equations become
Velocity is derived from potential theory and pressure is from linearized Bernoulli's equations.
A solution formula_6 is a transversal wave if the pressure formula_7 is identically zero and the velocity field is solenoidal. The equations are
Then the complete Oseen solution is given by
a splitting theorem due to Horace Lamb. The splitting is unique if conditions at infinity (say formula_10) are specified.
For certain Oseen flows, further splitting of transversal wave into irrotational and rotational component is possible formula_11 Let formula_12 be the scalar function which satisfies formula_13 and vanishes at infinity and conversely let formula_14 be given such that formula_15, then the transversal wave is
where formula_12 is determined from formula_18 and formula_19 is the unit vector. Neither formula_20 or formula_21 are transversal by itself, but formula_22 is transversal. Therefore,
The only rotational component is being formula_21.
The fundamental solution due to a singular point force embedded in an Oseen flow is the Oseenlet. The closed-form fundamental solutions for the generalized unsteady Stokes and Oseen flows associated with arbitrary time-dependent translational and rotational motions have been derived for the Newtonian and micropolar fluids.
Using the Oseen equation, Horace Lamb was able to derive improved expressions for the viscous flow around a sphere in 1911, improving on Stokes law towards somewhat higher Reynolds numbers. Also, Lamb derived—for the first time—a solution for the viscous flow around a circular cylinder.
The solution to the response of a singular force formula_25 when no external boundaries are present be written as
If formula_27, where formula_28 is the singular force concentrated at the point formula_29 and formula_30 is an arbitrary point and formula_31 is the given vector, which gives the direction of the singular force, then in the absence of boundaries, the velocity and pressure is derived from the fundamental tensor formula_32 and the fundamental vector formula_33
Now if formula_25 is arbitrary function of space, the solution for an unbounded domain is
where formula_37 is the infinitesimal volume/area element around the point formula_29.
Without loss of generality formula_39 taken at the origin and formula_40. Then the fundamental tensor and vector are
where formula_43 is the modified Bessel function of the second kind of order zero.
Without loss of generality formula_44 taken at the origin and formula_45. Then the fundamental tensor and vector are
Oseen considered the sphere to be stationary and the fluid to be flowing with a flow velocity (formula_48) at an infinite distance from the sphere. Inertial terms were neglected in Stokes’ calculations. It is a limiting solution when the Reynolds number tends to zero. When the Reynolds number is small and finite, such as 0.1, correction for the inertial term is needed. Oseen substituted the following flow velocity values into the Navier-Stokes equations.
Inserting these into the Navier-Stokes equations and neglecting the quadratic terms in the primed quantities leads to the derivation of Oseen's approximation:
Since the motion is symmetric with respect to formula_51 axis and the divergence of the vorticity vector is always zero we get:
the function formula_53 can be eliminated by adding to a suitable function in formula_51, is the vorticity function, and the previous function can be written as:
and by some integration the solution for formula_12 is:
thus by letting formula_51 be the "privileged direction" it produces:
then by applying the three boundary conditions we obtain
the new improved drag coefficient now become:
and finally, when Stokes' solution was solved on the basis of Oseen's approximation, it showed that the resultant drag force is given by
The force from Oseen's equation differs from that of Stokes by a factor of
In the far field formula_71 ≫ 1, the viscous stress is dominated by the last term. That is:
The inertia term is dominated by the term:
The error is then given by the ratio:
This becomes unbounded for formula_71 ≫ 1, therefore the inertia cannot be ignored in the far field. By taking the curl, Stokes equation gives formula_76 Since the body is a source of vorticity, formula_77 would become unbounded logarithmically for large formula_78 This is certainly unphysical and is known as Stokes' paradox.