text stringlengths 13 991 |
|---|
one arrives to the Crocco–Vazsonyi form (Crocco, 1937) of the Euler momentum equation: |
In the steady case the two variables entropy and total enthalpy are particularly useful since Euler equations can be recast into the Crocco's form: |
Finally if the flow is also isothermal: |
by defining the specific total Gibbs free energy: |
the Crocco's form can be reduced to: |
From these relationships one deduces that the specific total free energy is uniform in a steady, irrotational, isothermal, isoentropic, inviscid flow. |
Shock propagation is studied – among many other fields – in aerodynamics and rocket propulsion, where sufficiently fast flows occur. |
To properly compute the continuum quantities in discontinuous zones (for example shock waves or boundary layers) from the "local" forms (all the above forms are local forms, since the variables being described are typical of one point in the space considered, i.e. they are "local variables") of Euler equations through finite difference methods generally too many space points and time steps would be necessary for the memory of computers now and in the near future. In these cases it is mandatory to avoid the local forms of the conservation equations, passing some weak forms, like the finite volume one. |
Starting from the simplest case, one consider a steady free conservation equation in conservation form in the space domain: |
where in general F is the flux matrix. By integrating this local equation over a fixed volume Vm, it becomes: |
Then, basing on the divergence theorem, we can transform this integral in a boundary integral of the flux: |
This "global form" simply states that there is no net flux of a conserved quantity passing through a region in the case steady and without source. In 1D the volume reduces to an interval, its boundary being its extrema, then the divergence theorem reduces to the fundamental theorem of calculus: |
that is the simple finite difference equation, known as the "jump relation": |
Or, if one performs an indefinite integral: |
On the other hand, a transient conservation equation: |
For one-dimensional Euler equations the conservation variables and the flux are the vectors: |
In the one dimensional case the correspondent jump relations, called the Rankine–Hugoniot equations, are:< |
In the steady one dimensional case the become simply: |
Thanks to the mass difference equation, the energy difference equation can be simplified without any restriction: |
where formula_174 is the specific total enthalpy. |
These are the usually expressed in the convective variables: |
The energy equation is an integral form of the Bernoulli equation in the compressible case. |
The former mass and momentum equations by substitution lead to the Rayleigh equation: |
Since the second term is a constant, the Rayleigh equation always describes a simple line in the pressure volume plane not depending of any equation of state, i.e. the Rayleigh line. By substitution in the Rankine–Hugoniot equations, that can be also made explicit as: |
One can also obtain the kinetic equation and to the Hugoniot equation. The analytical passages are not shown here for brevity. |
The Hugoniot equation, coupled with the fundamental equation of state of the material: |
describes in general in the pressure volume plane a curve passing by the conditions (v0, p0), i.e. the Hugoniot curve, whose shape strongly depends on the type of material considered. |
It is also customary to define a "Hugoniot function": |
allowing to quantify deviations from the Hugoniot equation, similarly to the previous definition of the "hydraulic head", useful for the deviations from the Bernoulli equation. |
On the other hand, by integrating a generic conservation equation: |
on a fixed volume Vm, and then basing on the divergence theorem, it becomes: |
By integrating this equation also over a time interval: |
Now by defining the node conserved quantity: |
In particular, for Euler equations, once the conserved quantities have been determined, the convective variables are deduced by back substitution: |
Then the explicit finite volume expressions of the original convective variables are:< |
\oint_{\partial V_m}\rho\mathbf{u} \cdot \hat{n}\, ds\, dt \\[1.2ex] |
\mathbf u_{m,n+1} &= \mathbf u_{m,n} - \frac{1}{\rho_{m,n} V_m}\int_{t_n}^{t_{n+1}}\oint_{\partial V_m} (\rho\mathbf{u} \otimes \mathbf{u} - p\mathbf{I}) \cdot \hat{n}\, ds\, dt \\[1.2ex] |
\mathbf e_{m,n+1} &= \mathbf e_{m,n} - \frac{1}{2}\left(u^2_{m,n+1} - u^2_{m,n}\right) - \frac{1}{\rho_{m,n} V_m}\int_{t_n}^{t_{n+1}}\oint_{\partial V_m} \left(\rho e + \frac{1}{2}\rho u^2 + p\right)\mathbf{u} \cdot \hat{n}\, ds\, dt \\[1.2ex] |
It has been shown that Euler equations are not a complete set of equations, but they require some additional constraints to admit a unique solution: these are the equation of state of the material considered. To be consistent with thermodynamics these equations of state should satisfy the two laws of thermodynamics. On the other hand, by definition non-equilibrium system are described by laws lying outside these laws. In the following we list some very simple equations of state and the corresponding influence on Euler equations. |
For an ideal polytropic gas the fundamental equation of state is: |
where formula_53 is the specific energy, formula_69 is the specific volume, formula_71 is the specific entropy, formula_193 is the molecular mass, formula_194 here is considered a constant (polytropic process), and can be shown to correspond to the heat capacity ratio. This equation can be shown to be consistent with the usual equations of state employed by thermodynamics. |
From this equation one can derive the equation for pressure by its thermodynamic definition: |
By inverting it one arrives to the mechanical equation of state: |
Then for an ideal gas the compressible Euler equations can be simply expressed in the "mechanical" or "primitive variables" specific volume, flow velocity and pressure, by taking the set of the equations for a thermodynamic system and modifying the energy equation into a pressure equation through this mechanical equation of state. At last, in convective form they result: |
and in one-dimensional quasilinear form they results: |
In the case of steady flow, it is convenient to choose the Frenet–Serret frame along a streamline as the coordinate system for describing the steady momentum Euler equation: |
where formula_1, formula_15 and formula_40 denote the flow velocity, the pressure and the density, respectively. |
Let formula_204 be a Frenet–Serret orthonormal basis which consists of a tangential unit vector, a normal unit vector, and a binormal unit vector to the streamline, respectively. Since a streamline is a curve that is tangent to the velocity vector of the flow, the left-hand side of the above equation, the convective derivative of velocity, can be described as follows: |
where formula_206 is the radius of curvature of the streamline. |
Therefore, the momentum part of the Euler equations for a steady flow is found to have a simple form: |
For barotropic flow formula_208, Bernoulli's equation is derived from the first equation: |
The second equation expresses that, in the case the streamline is curved, there should exist a pressure gradient normal to the streamline because the centripetal acceleration of the fluid parcel is only generated by the normal pressure gradient. |
The third equation expresses that pressure is constant along the binormal axis. |
Let formula_210 be the distance from the center of curvature of the streamline, then the second equation is written as follows: |
In a steady flow of an inviscid fluid without external forces, the center of curvature of the streamline lies in the direction of decreasing radial pressure. |
Although this relationship between the pressure field and flow curvature is very useful, it doesn't have a name in the English-language scientific literature. Japanese fluid-dynamicists call the relationship the "Streamline curvature theorem". |
This "theorem" explains clearly why there are such low pressures in the centre of vortices, which consist of concentric circles of streamlines. |
This also is a way to intuitively explain why airfoils generate lift forces. |
All potential flow solutions are also solutions of the Euler equations, and in particular the incompressible Euler equations when the potential is harmonic. |
Solutions to the Euler equations with vorticity are: |
In fluid dynamics the Milne-Thomson circle theorem or the circle theorem is a statement giving a new stream function for a fluid flow when a cylinder is placed into that flow. It was named after the English mathematician L. M. Milne-Thomson. |
Let formula_1 be the complex potential for a fluid flow, where all singularities of formula_1 lie in formula_3. If a circle formula_4 is placed into that flow, the complex potential for the new flow is given by |
with same singularities as formula_1 in formula_3 and formula_4 is a streamline. On the circle formula_4, formula_10, therefore |
Consider a uniform irrotational flow formula_12 with velocity formula_13 flowing in the positive formula_14 direction and place an infinitely long cylinder of radius formula_15 in the flow with the center of the cylinder at the origin. Then formula_16, hence using circle theorem, |
represents the complex potential of uniform flow over a cylinder. |
The Prony equation (named after Gaspard de Prony) is a historically important equation in hydraulics, used to calculate the head loss due to friction within a given run of pipe. It is an empirical equation developed by Frenchman Gaspard de Prony in the 19th century: |
where "hf" is the head loss due to friction, calculated from: the ratio of the length to diameter of the pipe "L/D", the velocity of the flow "V", and two empirical factors "a" and "b" to account for friction. |
This equation has been supplanted in modern hydraulics by the Darcy–Weisbach equation, which used it as a starting point. |
In fluid dynamics, the Darcy friction factor formulae are equations that allow the calculation of the Darcy friction factor, a dimensionless quantity used in the Darcy–Weisbach equation, for the description of friction losses in pipe flow as well as open-channel flow. |
The Darcy friction factor is also known as the "Darcy–Weisbach friction factor", "resistance coefficient" or simply "friction factor"; by definition it is four times larger than the Fanning friction factor. |
In this article, the following conventions and definitions are to be understood: |
Which friction factor formula may be applicable depends upon the type of flow that exists: |
Transition (neither fully laminar nor fully turbulent) flow occurs in the range of Reynolds numbers between 2300 and 4000. The value of the Darcy friction factor is subject to large uncertainties in this flow regime. |
The Blasius correlation is the simplest equation for computing the Darcy friction |
factor. Because the Blasius correlation has no term for pipe roughness, it |
is valid only to smooth pipes. However, the Blasius correlation is sometimes |
used in rough pipes because of its simplicity. The Blasius correlation is valid |
The Darcy friction factor for fully turbulent flow (Reynolds number greater than 4000) in rough conduits can be modeled by the Colebrook–White equation. |
The last formula in the "Colebrook equation" section of this article is for free surface flow. The approximations elsewhere in this article are not applicable for this type of flow. |
Before choosing a formula it is worth knowing that in the paper on the Moody chart, Moody stated the accuracy is about ±5% for smooth pipes and ±10% for rough pipes. If more than one formula is applicable in the flow regime under consideration, the choice of formula may be influenced by one or more of the following: |
The phenomenological Colebrook–White equation (or Colebrook equation) expresses the Darcy friction factor "f" as a function of Reynolds number Re and pipe relative roughness ε / "D"h, fitting the data of experimental studies of turbulent flow in smooth and rough pipes. |
The equation can be used to (iteratively) solve for the Darcy–Weisbach friction factor "f". |
For a conduit flowing completely full of fluid at Reynolds numbers greater than 4000, it is expressed as: |
Note: Some sources use a constant of 3.71 in the denominator for the roughness term in the first equation above. |
The Colebrook equation is usually solved numerically due to its implicit nature. Recently, the Lambert W function has been employed to obtain explicit reformulation of the Colebrook equation. |
Additional, mathematically equivalent forms of the Colebrook equation are: |
The additional equivalent forms above assume that the constants 3.7 and 2.51 in the formula at the top of this section are exact. The constants are probably values which were rounded by Colebrook during his curve fitting; but they are effectively treated as exact when comparing (to several decimal places) results from explicit formulae (such as those found elsewhere in this article) to the friction factor computed via Colebrook's implicit equation. |
Equations similar to the additional forms above (with the constants rounded to fewer decimal places, or perhaps shifted slightly to minimize overall rounding errors) may be found in various references. It may be helpful to note that they are essentially the same equation. |
Another form of the Colebrook-White equation exists for free surfaces. Such a condition may exist in a pipe that is flowing partially full of fluid. For free surface flow: |
The above equation is valid only for turbulent flow. Another approach for estimating "f" in free surface flows, which is valid under all the flow regimes (laminar, transition and turbulent) is the following: |
where "Reh" is Reynolds number where "h" is the characteristic hydraulic length (hydraulic radius for 1D flows or water depth for 2D flows) and "Rh" is the hydraulic radius (for 1D flows) or the water depth (for 2D flows). The Lambert W function can be calculated as follows: |
The "Haaland equation" was proposed in 1983 by Professor S.E. Haaland of the Norwegian Institute of Technology. It is used to solve directly for the Darcy–Weisbach friction factor "f" for a full-flowing circular pipe. It is an approximation of the implicit Colebrook–White equation, but the discrepancy from experimental data is well within the accuracy of the data. |
The Swamee–Jain equation is used to solve directly for the Darcy–Weisbach friction factor "f" for a full-flowing circular pipe. It is an approximation of the implicit Colebrook–White equation. |
Serghides's solution is used to solve directly for the Darcy–Weisbach friction factor "f" for a full-flowing circular pipe. It is an approximation of the implicit Colebrook–White equation. It was derived using Steffensen's method. |
The solution involves calculating three intermediate values and then substituting those values into a final equation. |
The equation was found to match the Colebrook–White equation within 0.0023% for a test set with a 70-point matrix consisting of ten relative roughness values (in the range 0.00004 to 0.05) by seven Reynolds numbers (2500 to 108). |
Goudar equation is the most accurate approximation to solve directly for the Darcy–Weisbach friction factor "f" for a full-flowing circular pipe. It is an approximation of the implicit Colebrook–White equation. Equation has the following form |
Brkić shows one approximation of the Colebrook equation based on the Lambert W-function |
The equation was found to match the Colebrook–White equation within 3.15%. |
Brkić and Praks show one approximation of the Colebrook equation based on the Wright formula_40-function, a cognate of the Lambert W-function |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.