## 2009年11月10日 (火)

### New viscous fluid calculation and the existence of Non-Flow-Layer

from Hiroshima Hiro. Hiroo Oyama
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I'll discuss about the natural convection, in which we can't neglect the viscosity. It occurs in a lamp within an electric bulb, or it occurs in a meteorological phenomena. It takes a long time to analyze the natural flow phenomena by the computer simulation.
Heat-Fluid-Analysis dominated by buoyancy is a typical case. Partly departed the general method to solve it with the NAVIER-STOCKES' equation, I started to think about more convenient method.
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In this paper, I'll introduce a method to solve conveniently these natural flow problems, especially dominated by buoyancy and viscosity. This method is in relation to the Poiseuille's law, used in the experiment to measure viscosity. But it shorten calculative time like an 1/100 or 1/10,000. This method has an assumption of non-flow-layer, which lays in the boundary near solids' surface, and has the calculations of local exchanging of heat fluid by the difference of buoyancy. This assumption bridges between the Limit of computer-simulation in which the space is departed only by finite division and the differential equation of continual fluid which is under the infinite division mathematically. Using this method, we can be free from the heat-translate-ratio and become to be able to do computation with speed and rationality.
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Fig1(A) is the measured data(1) by Kondou, which is concerned to Logarithmic law. Its general formula is
U = (U~/κ)*Ln(Z/Z0)    　　・・・ (1)
,where U~; friction velocity{ =√(τ/ρ）},  κ ; Karman's constant, Z0 ; roughness height. Z ; observing height, U ; velocity on observing height.
If we assume that this equation is true in the nearest region from the ground, we can get Fig.1(B) under the condition (U~/κ) = 1m/s,  Z0 = 1.2cm.
Though this is a famous experimental equation, it contains an assumption that the air in some layer doesn't move substantially. We call it Non-Flow-Layer. Paying attention to the existence of this layer, I propose the new method( CURL ) for high speed heat-fluid-analysis.
The definition of CURL
There are 4 pieces of fluid element closed each other in Fig.2(A). This is the smallest model of natural convection, which is able to be happened by buoyancy. By 4 buoyancy(F1,F2,F3,F4), a rotational force is occurred in the center. The force C(the capital Letter of CURL) has the direction( counterclockwise : positive ), and defined by the following formula.
C ≡ A*( F1-F2-F3+F4 )*L^3   ・・・ (2)
, where L is a side of small piece element, L^3 is the volume of one. Buoyancy(F1) has a unit of pressure force per volume, and A is a constant defined after. Except the C of the center, there are 8-CURLs, which are showed by sign "C". Each of them includes one or more solid matter, so C=ZERO is set automatically. Then, a half of the side (L/2) becomes to be defined as substantial Non-Flow-Layer.
Fig.2(B) shows "There are 6 small pieces of fluid matter, and they make influences with each other by buoyancy." For the fluid's moving volume, I want to suppose to be proportional to small time ⊿ｔ, be proportional to the Difference of CURLs[C1-C2], be inverse-proportional to viscosity μ. A'/(4π^2) is a proportional constant which is described after. Then,
D ≡ A'/(4π^2)*[C2-C1] ・⊿t/μ　  ・・・ (3)
Exchanged A in (2) with { A'/(4π^2)・⊿t/μ }, then,
C ≡ { A'/(4π^2)・⊿t/μ・L^3 }(F1-F2-F3+F4)　・・（2'）
and,  D ≡ C2－C1　　　　　　　　　　　 ・・・（3'）
We can set 16-CURLs as C11～C44 in Fig.2(C), all except 3-CURLs around the center are ZERO. Through this treatment, for a composited flow as the arrows come up, which pattern, of course, changes according to the temperature distribution.
In Fig.3(A), there are two standing wall. When the gradient of temperature exists, a convection is occurred. According to the upper definition, nearest elements have the non-flow-layer(see C2: consists of A,B,C,D).
But this discussion is only concerned about the elements lined perpendicular to the wall, and is only concerned about the elements parallel to the Wall ( see C3 : of C,D,E,F, C1 : of G,H,J,K ).  So C3≠ZERO,　C1≠ZERO. But C2=ZERO, because 2 elements(A,B) are inside of the wall.
Furthermore, Fig.3(B) is the case that several elements of wall are Lined diagonally, and the Non-Flow-Layer is happened as slant direction by the vector transaction. On a case that a mosaic model is set by the actual body's 3-Dimensional-surface, the surface of Non-Flow-layer happens to be like an off-set body's surface.
Simulation examples.
Fig.4(A) shows the finally determined flow. It shows the convection phenomena very nicely, especially in the nearest region of bulb, on which air moves bundling up the bulb, and the distribution of arrows of flow.
(B) shows its temperature distribution. Nevertheless of simple and rough model, it has an reality. The center numeric of each square is the calclating result, and contour lines are drawn accurately from the numeric distribution. (A) and (B) are profiles. As this calculation was done in 3-definishion, any section's distribution can be drawn.
Generally speaking, there are three heat transmission types( radiation, convection, conduction). If the balance of three intensities is not suit to real world, the temperature's distribution become to be different far from the measured.
Fig.5 is the typical result of cylindrical lamp. Compared  the measured values ( X ) with the calculated curves, we can realize that they have a good coincidence and three intensities are treated in a good balance.
Flow and Sequence of the calculation.
(1) At first, Calculate the buoyancy distribution
[ Fijk = Tijk/T0 -1 ]
Triple Do-loop[X,Y,Z] is set in program, and
(2) Calculate CURL is memorized in CURL-DIMENSION-MEMORY one after another.
(3) D-CURL is calculated as the difference of CURLs.
(4) Subtract the heat-out from the element, and add the heat-in to it, and continue to calculate the distribution which is just after ⊿ｔ.
(5) Using these result, the upper (1)～(4) is repeated for the next ⊿t.
Compared to the general method with simultaneous equations, this method gives a very high speed calculation for you. The reason comes from that formula and calculation logic are extremely simple.
Coincidence about CURL and P's Law.
When we estimate the motion of fluid, we usually use the Absolute coordinates and the Relative coordinates. The former is the outlook of the observing phenomena, and the observing object, for example a flying-ball, moves on the screen. The latter moves with the observing object, and the ball can be seen at the center of screen and only the around air moves on it.
Here at first, I'll describe about a Local coordinate, which belongs to the Relative. We make small cut pieces of observing space, and discuss about the average motion of substance in each cut piece.  ⊿t is a divided small time. the average velocity is called Vi. Here V0 is the vector of velocity which is averaged in nearest around area. In this ⊿t, suppose a corollary that moves with the vector V0. This is the local coordinates. If N is the number of cut pieces of observing space, there are N of local coordinates.
In the CURL method, we exchange and use the two corollary of calculation in every each ⊿t. Inertial force in this computation should be counted in only the absolute coordinates, and in the local coordinates it becomes to be nonsense. From the viewpoint of vector computation, Fluid path line is calculated as the total vector distribution, which is assembled with the distribution in local coordinates. In every each ⊿t, You can connect each vectors of local coordinates without contradicting to the law of continuation.
The motion happened by buoyancy is the changing phenomena in the nearest region of observing point. The many local changing motions are combined, totalized without contradiction of the distribution of solid's element and of the temperature distribution of substance, and final stream line (exactly, path line) becomes to be appeared. For that reason that inertial force can be neglected on the each local coordinates, in each local coordinates in ⊿t, inertial force becomes to be negligible in NAVIER-STOKES' equation. This makes it very easy for the analysis of phenomena in the boundary layer.
There is Hargen-Poiseuille's law concerning about the flow volume through a tube in ⊿t(sec.). ( G.Hagen,1839 ; J.Poiseuille,1840 )
V = (π/8)・(r^4・⊿P・⊿t)/(μ・h)　　・・・(4)
where, V : the flow volume( cc ) in ⊿t, r : tube's radius( ㎝ ), h : tube's length（ ㎝ ）, μ : viscosity( gr./(㎝・sec) ), ⊿P : difference of pressure( dyne/(㎝・㎝）=ｇｒ/（㎝・sec・sec).
As this equation has a good coincidence to the experiment, it is often used as the method of viscosity measurement. That is why it is very famous. With this law, I leaded the equation how the flow volume should be estimated in the local coordinates where inertial force is negligible. Avoiding the details( ref. Fig.6 ), I'll describe the equation.
D = A"/(4π^2)/・L^3・[ F23+F11-F13+F21 ]・⊿t/μ ・・・(5)
With C2 = (F23-F13-F12+F22)*L^3　and  C1 = (F22-F12-F11+F21)*L^3
the formula(3)  D ≡ A'/(4π^2)*[C2-C1] ・⊿t/μ ・・・ (3)
becomes to be
D = A'/(4π^2)*[F23-F13+F11-F21]*L^3・⊿t/μ ・・・（３')
The constant A" is about 1, and we can set as A'≒A"≒１.
⇒　(5) = (３')
So we can use the CURL-method as if adapting Poiseuille's law.
Discussion
Navier-Stokes' equation needs a lot of time and complicated calculations. This equation is perfect mathematically. But the practical division should be done and the practical model should be prepared. For me, it seems to be happened the unbalance between the accuracy of differential equation and the accuracy of mosaic modeling actually. CURL's theory itself is stepped with a moderate modeling. As the result, you can compute the flow with a moderate accuracy and with highest speed. You can be free from the Heat-translate-ratio, and CURL's method may have a value to be examined only for the convenience of first condition's setting.
The boundary layer is defined as "1% less velocity range compared with the velocity of far from a solid surface." From this definition you can say that whole natural fluid phenomena is included in the boundary layer, and all observing space of phenomena is in it. In the nearest region filament coil in an electric bulb, Langmurer-sheath is known, which does not move substantially. It may be called Non-flow-layer. The CURL method is a theory that the layer like Langmuer-sheath exists in other flow phenomena.
In the atmospheric boundary layer, Canopy-Layer is defined. It is explained to be a thin layer in which obstacles( buildings, trees, houses, etc.) are existed just on the ground, and a wind has a very different velocity and different direction in it, compared with the sufficiently upper wind. The wind in Canopy-Layer often blows to the opposite direction.
We'll divide the air and the shallow ground into small cubes which each side has the double length of average Canopy-Layer. Suppose vi ; vector of average moving velocity in an i-element, Vi ; vector of sufficiently upper wind. Then |vi| << |Vi| !　 When we discuss about an atmospheric natural convection including the influence of ground shape, I think it reasonable to treat the Canopy-Layer for Non-Flow-Layer.
In the simulation of wings of airplane, we have to treat the air for a compressible fluid, and adiabatic change should be considered. In the shock wave which is happened around the wings non-reversible phenomena is happened like a viscous exothermic phenomena or heat conduction. As heat does go in and out locally, CURL's method in local coordinates seems to be favourable for it.
There is D'Alambert's Paradox which means vortex never vanish forever. This paradox happens to be by the ignorance of viscosity. The CURL of this letter was proposed as a hypothesis to supplement the limit that space can be divided only by finite number in mathematical fluid calculation.
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References
1) Junsei Kondou : Introduce Meteorological Science, 1987, Tokyo-Daigaku-Shuppankai
2) Norihiko Sumitani : Continual fluid dynamics, 1969, Kyouritu-Shuppan-Sha
3) Takesuke Fujimoto : Fluid dynamics, 1970, Youken-Dou
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Fig.1 Logarithmic Law  Fig.2 Definition of CURL & D-CURL Fig.3 CURL's occurrence on the nearest region around wall. Fig.4 Examples of computer simulation. Fig.5 When you turn the bulb, then the distribution changes. Fig.6 In a local region, DCURL(Differential-CURL)=CURL2-CURL1

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