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УДК 532.526.4
ТЕОРИЯ ТУРБУЛЕНТНОСТИ И МОДЕЛИРОВАНИЕ ТУРБУЛЕНТНОГО ПЕРЕНОСА В АТМОСФЕРЕ ЧАСТЬ З
Трунев Александр Петрович к. ф.-м. н., Ph.D.
Директор, A&E Trounev IT Consulting, Торонто, Канада
В работе представлена полностью замкнутая модель турбулентного пограничного слоя, полученная из уравнения Навье-Стокса. Фундаментальные константы пристенной турбулентности, включая постоянную Кармана, определены из теории. Эта модель была развита для ускоренного и неизотермического пограничного слоя над шероховатой поверхностью
Ключевые слова: ТУРБУЛЕНТНЫЙ ПЕРЕНОС, УСКОРЕННЫЕ ТЕЧЕНИЯ, ПОГРАНИЧНЫЙ СЛОЙ, ШЕРОХОВАТАЯ ПОВЕРХНОСТЬ, ПРИЗЕМНЫЙ СЛОЙ АТМОСФЕРЫ
UDC 532.526.4
THEORY OF TURBULENCE AND SIMULATION OF TURBULENT TRANSPORT IN THE ATMOSPHERE PART 3
Alexander Trunev Ph.D.
Director, A&E Trounev IT Consulting, Toronto, Canada
The completely closed model of wall turbulence was derived directly from the Navier-Stokes equation. The fundamental constants of wall turbulence including the Karman constant have been calculated within a theory. This model has been developed also for the accelerated and non-isothermal turbulent boundary layer flows over rough surface
Keywords: TURBULENT TRANSPORT, ACCELERATED FLOW, BOUNDARY LAYER, ROUGH SURFACE, ATMOSPHERIC SURFASE LAYER
3 Model of turbulent flows over rough surface
3.1 Empirical models of turbulent flow over rough surfaces
The study of the rough wall turbulence is important in fluid mechanics, in the atmosphere and ocean and in engineering flows [1-74]. The roughness effect on the turbulent boundary layer have been considered and summarised by Niku-radse [75] Schlichting [61,76], Bettermann [77], Millionschikov [78], Dvorak [79], Dirling [80], Simpson [81], Dalle Donne & Meyer [82] and other.
Nikuradse [75] established (for sand-roughened pipes) that if the roughness height significantly exceeds the viscous sublayer thickness, then the mean velocity profile can be described by the logarithmic function:
U 1 z N
u;=k\+c- (3.1)
where u, is the friction velocity, u, =^Jt / p, t is the wall shear stress, p is the fluid density, z is the distance from the wall - see Figure 3.1, ks is the characteristic scale of the sand roughness, k,cs are empirical values. Nikuradse found that k = 0.4,cs = 8.5 for the completely rough regime. He compared the mean velocity profile (3.1) with the law of the wall, derived him before in 1932 for turbulent flows in smooth pipes, as follows
— = lin^+co (3.2)
u, k v u.
v is the kinematic viscosity, к = 0.4, c0 = 5.5 are the logarithmic profile constants for the hydraulically smooth surface. DU is the shift of the mean velocity logarithmic profile which can be defined for the turbulent boundary layer over a rough surface as
DU = ±,n HA+Ds (3.3)
ut к v s
Ds » -3 0 for the completely rough regime. Nikuradse has shown that the dimensionless roughness height parameter ks + = uxks / v can be used as an indicator of the rough wall turbulence regime. He proposed to consider three typical cases:
■ the hydraulically smooth wall for 0 < ks + < 5, DU = 0;
■ the transitionally rough regime for 5 < ks + < 70, Ds varies with ks +;
■ the completely rough regime for ks + > 70, Ds » -3 0.
Thus, the sand-roughened wall turbulence depends on the dimensionless roughness height (roughness Reynolds number) ks+ as have been established by Nikuradse.
Schlichting [76], used the Nikuradze's date base and his own experimental results obtained in the water tunnel of rectangular cross section with the upper rough wall, proposed the new form of the equation (3.1) which is well counted the roughness effect on the turbulent boundary layer by means of the effective wall location (Dz) and the equivalent sand roughness parameter kes. With these parameters the mean velocity profile in the turbulent flow over an arbitrary
rough surface can be written in the Nikuradze's form (3.1) as follows:
U + c. (3.4)
ut к kes
where Zj = z-Dz (see Figure 3.1). The effective wall location was defined by Schlichting as the mean height of the roughness elements (the location of a "smooth wall that replaces the rough wall in such a manner as to keep the fluid volume the same"). The value kes has been measured by Schlichting for the several types of the roughness elements with various shapes, sizes and spacing.
z = S0:u = U0 z, z = h(x,y,t) ЛЩЛ /ЯЧ /fflv-z fa Z1 ■i—Л ± kr i_ 3D k kr/§\
VdV + l^-d d f> -t>l +-t>l d 1<k 2D
X *- L —!>
Figure 3.1: The scheme of the turbulent flow over a rough surface (left) and the roughness element geometry (right): spheres, spherical segments, conical elements (3D) and transverse rectangular roods (2D)
The Schlichting's experiment was re-evaluated by Coleman et. al. [83] and they noticed that some Schlichting's data have been obtained in the transitional rough regime.
Clauser [84] has shown that the shift of the mean velocity profile can be written as
^ = -Lin Ut + D
U, K V
where kr is the characteristic scale of roughness elements and D must be some function of the roughness geometrical parameters. Hence the equivalent sand roughness parameter kes = kr exp[K (D - Ds)], where Ds «-3.0 for sand roughness.
Bettermann [77] discovered that D is the function of the roughness elements spacing. He introduced the roughness density parameter for roughness composed of the transverse square bars as the pith-to-height ratio, AB = L / kr - see
Figure 3.1. Bettermann found that in the range 1 <AB < 5 the variations of D
with the roughness density can be specified by
D = 12.25lnAB - 17.35
As has been demonstrated by Dvorak [79], the rough wall effect well correlated with the roughness density parameter defined as pitch-to-width ratio or the ratio of total surface area to roughness area, A s = L / d. Dvorak developed the Bettermann's model in the range 4.68 < As < 102, used the data of Schlichting and other researches, as follows:
[12.25ln As -17.35, 1 <As < 4.68
D = \ s s (3 5)
[- 2.85ln As + 5.95, As > 4.68 v ' J
Simpson [81] introduced the roughness density parameter in the case of three-dimensional (3D) roughness as A*s = (NSAF)-1 where NS is the number of
significant roughness elements per unit area, AF is the average frontal area of the significant roughness elements. He suggested the general interpretation of the Bettermann-Dvorak correlation (3.5): two branches (3.5) exist depending on the formation or absence of transverse vortices between roughness elements. Simpson also showed that the shape of the element is an important parameter.
The model been reported by Dirling [80] and verified by Grabow & White [85], takes into consideration the roughness elements shape parameters. The Dirling's density parameter is defined as AD = (L / kr)(AW / AF)4 3 where AW is "the windward wetted surface area". In a case of two-dimensional (2D) roughness the Dirling's parameter leads to the Bettermann's roughness density parameter. As it was shown by Sigal & Danberg [86] the shape parameters effect can be described by the similar correlation such the equation (3.5) and that D = 2.2 for the two-dimensional roughness in the range 4.89 < As < 13.25. They also underlined that the correlation for 2D roughness elements is not the same as for 3D elements. On the other hand, Kind & Lawrysyn [87] confirmed that the Better-mann-Dvorak function D( A s) in the form (3.5) can be successfully used for the correlation of experimental data in the aerodynamic experiments with the natural hoar-frost roughness.
Dalle Donne & Meyer [82] correlated their data and those of previously researches (data bases [88-105] considered below) used the roughness density parameter A*d = (L - d) / kr. They developed the empirical model which can be applied to the turbulent flows in the annuli and tubes with inner surface roughened by rectangular ribs.
The roughness density parameter entered by Dalle Donne & Meyer [82] in case of 2D roughness elements can be transformed as follows
With this parameter the experimental data of Dalle Donne & Meyer [82] and other sources [88-105] summarized in Table 3.1 can be described by equations:
This correlation has been derived by Dalle Donne & Meyer [82] for the range of the experimental data parameters 0.086 < kr / d < 5.0 and 1.85 < As < 980 . Therefore the rough surface effect depends on two roughness parameters kr / d and A*d . Thus there is no any "universal" parameter for 2D roughness elements in the common case. But the experimental data with various kr / d can be plotted together as the graph of the function
л; = (L - d)/kr = (d / kr)(Л, -1)
к
;(Л*; ) = C0 + (2 + 7/ Л*; )lg d - R,
d
(3.6)
; (Л*;) = ;Л*;) - (2 + 7/ Л*;) lg(кг / d) .
Table 3.1. Geometrical characteristic of 2D roughness investigated by various authors
Authors Year Geometry L / d kr / d Symbol
Mobius 1940 Tube 10.0-29.22 0.3-2.20 3
Chu & Streeter 1949 Tube 1.95-7.57 0.93 4
Sams 1952 Tube 2.0-2.3 0.88-1.37 9
Nunner 1956 Tube 16.36 0.8 16
Koch 1958 Tube 9.8-980 1.0-5.0 5
Fedynskii 1959 Annulus 6.67-16.7 1.0 10
Draycott & Lawther 1961 Annulus 2.0 1.0 2
Skupinski 1961 Annulus Tube 2.0-41.0 22.2-133.4 1.0 2.0 6
Savage & Myers 1963 Tube 3.66-43.72 1.33-2.67 13
Perry & Joubert 1963 Wind tunnel 4.0 1.0 19
Sheriff, Gumley & France 1963 Annulus 2.0-10.0 1.0 14
Gargaud & Paumard 1964 Tube Annulus 1.8-16.0 10.0-16.0 1.0-1.67 1.0 1
Bettermann 1966 Wind tunnel 2.65-4.18 1.0 20
Massey 1966 Annulus 7.53-30.15 1.06 15
Kjellstrom & Larson 1967 Annulus 2.02-38.52 0.086-4.08 12
Fuerstein & Rampf 1969 Annulus 2.91-25.04 0.42-2.50 8
Lawn & Hamlin 1969 Annulus 7.61 1.0 17
Watson 1970 Annulus 6.49-7.22 1.0 11
Stephens 1970 Annulus 7.20 1.0 18
Webb, Eckert & Goldstein 1971 Tube 9.70-77.63 0.97-3.88 7
Antonia & Luxton 1971 Wind tunnel 4.0 1.0 21
Antonia & Wood 1975 Wind tunnel 2.0 1.0 22
Dalle Donne & Meyer 1977 Annulus 4.08-61.5 0.25-2.0 24
Pineau, Nguyen, Dickinson & Belanger 1987 Wind tunnel 4.0 1.0 23
Di
+S-t M . ^ 4-
+ +** . * /ъ X ^4* *M I н
* -1 4- -10
x -2 ♦ -11
/ ++
♦ 4/ + ■ У+ * \ » - 3 m - 4 + -12 * -13
/x++ * \ 2 \ 2 o-14
/ " M - 6 *.-15
* *-7 И -16
■ / " *-8 '*-17
—■ ■ ■ 1 —■ ■ ■ 1 *-9 *-18
1 10 100 л*
Figure 3.2: The rough surface effect on the turbulent flow: 2D roughness elements data [88-105], the solid line is calculated on the model of Dalle Donne & Meyer [82]
Figure 3.2 demonstrates D1( A*D) calculated according to (3.6) - solid line (1) and the experimental data found for 2D roughness elements by various authors listed in Table 3.1 (the corrected and reduced data or R(<* )01 from Table 2 of Dalle Donne & Meyer [82] has been used as long as correlation (3.6) was proposed for this values).The symbols description is given in the right part of Figure 3.2 and Table 3.1. As Figure 3.2 shows the correlation is good for the middle and high value of the roughness density parameter, but for L*D »1 the scatter of the points is rather large and can't be explained by the experimental technique differences only. The empirical model [82] can't explain the experimental behaviour of the mean velocity shift with a roughness density, which can be found out only by comparison of large number of the data, obtained by various authors [75-77, 81-83, 87-109].
Osaka & Mochizuki [110] examined d-type rough wall boundary layer in a transitionally and a fully rough regime. They have shown that in a transitionally rough regime the mean velocity logarithmic profile is confirmed and that the Karman constant has the same value as for the hydraulically smooth wall flow.
The mean velocity logarithmic profile widely used in the atmospheric turbulence research is given by (see [18-20, 25, 111-113]):
ut k z0
where zd is the displacement height, z0 is the roughness length. Note that zd and z0 are considered often as some adjustment parameters chosen for the best correlation of the local wind profile in the neutral stratified flow with the logarithmic profile. The model of the displacement height has been considered by
Jackson [112]. The classification of the experimentally determined roughness length for various terrain types was given by Wieringa [113].
3.2 Model of wall roughness effect on turbulent flow
The effective wall location was defined by Schlichting [76] as the mean height of the roughness elements and in the mathematical form can be written as:
Az = ra = L— JJ r(x, y)dxdy (3.7)
LxLy AxAy
where z = r(x,y)is the relief of the rough surface - see Figure 3.1, Lx,Ly are the rough wall scales in the x, y directions, AxAy = LxLy.
In a case of two dimensional roughness considered by Dvorak [79] and
Simpson [81] the roughness density parameter depends on the width and pitch of
roughness elements (see Figure 3.1): A s = L / d. The mean roughness height depends on the height of roughness elements as ra = akr / As, where a is the numerical constant which equals to unity in this case. Using the Bettermann-Dvorak's equation (3.5) in the rang 4.68 < As < 102 the shit of the mean velocity can be presented as a function of the mean roughness height, thus we have
— = !]n Utk- + D » 2.5 ln Utk- - 0.35ln A + 5.95 =
Ut k v n A,
u r
= 2.5ln—- 0.35ln A, + 5.95 v s
In this approach the mean velocity profile in the turbulent flow over a rough surface can be rewritten as follows
U 1 z
— = — l^^ + 0.35lnA - 0.45
U k ra
If we redefined the main roughness scale then the mean velocity profile takes the form which widely used in the atmosphere research:
U = -k-in ^ (3.8)
U k r0
where ln r0 = ln ra - 0.35kln As + 0.45k »ln ra - 0.14ln As + 0.18. Practically r0 » ra for As = 5 and r0 » 0 63ra for A s = 100. Hence, the logarithmic profile mainly depends on the mean height of the roughness elements in this range of the roughness density.
Let us consider the random function defined as
u(z, /r) = — ln— (3.9)
1 k r
where r is the random parameter with the mean value given by
ra = J0 rfs (r )dr
here fs = fs (r) is the density of a probability distribution function (roughness statistic function) normalised on unity:
J0 fs (r )dr = 1
Both parts of equation (3.9) can be averaged with this function as follows
U(z1) = J u(/ r)fs (r)dr = k J (in Zl - In r f f)dr = k In^
Л. - f p,
where lnr0 = J0 ln(r)fs(r)dr . With this result the mean-squared-value of the velocity fluctuations can be calculated as
5~2 = J(u -U)2 fs(r)dr = J(ln r - ln r0 )2 fs (r)dr
0 k 0
Therefore we have
Su2 = —L ^ ln2 A - ln2 r0)
Thus, the random function u(Zj / r) can be used for the mean velocity calculation as well as for the mean-squared-value of the velocity fluctuations modelling. Our main idea is that the random function ~(- / r) can be calculated on the basis of a solution of the Navier-Stokes equation due to the surface layer transformation
~(Zj /r) = lim Ju(x,y,h)dxdydz (3.10)
SV ® dVs SV
SV
where hi = - / r(x,y) is fixed over the integrated region, hi = -1/ r = const, SV is an arbitrary volume put in dV = LxLydz and containing dVs as a whole, dVs is the subregion in which altitude of the rough surface r(x, y) varies in limits from r up to r + dr , hence by definition dVs = dVfs (r)dr .
Note, that the surface layer transformation is only a kind of averaging procedure which conserves the function properties across a boundary layer. The Navier-Stokes equation can be averaged with the surface layer transformation
(3.10) instead the normal Reynolds averaging method to derive then the equation for the random amplitude ~(z1 / r). Unfortunately it's impossible to use this
0
method in the simple form (3.10), because, for example, in the case of a smooth flat plate r = 0 .
Therefore we suppose that there is a surface z = h(x,y, t) (the dynamic roughness surface) inside the flow domain which can be used for modelling the rough surface effect on the turbulent flow. Without any limits we can choose a surface z = h( x, y, t) close to the wall surface z = r (x, y), but not equal to r (x, y).
Let h(x,y,t) = r(x,y) + hr(x,y,t), where hr(x,y,t)is the height of the viscous sublayer over the rough surface. In the turbulent flow the surface z = h(x, y, t) can be described by random continuous parameters h, ht, hx, hy characterised the height, velocity and inclination of the surface elements. Let’s define the subregion dVs in which the local height of the rough surface r(x, y) varies in limits from r up to r + dr and parameters of the surface z = h(x, y, t) in limits from h up to h + dh , from ht up to ht + dht , from hx up to hx + dhx , from hy up to hy + dhy , thus
dVs = dVfs (r, h, hx, hy, ht )drdhdhxdhydht,
where fs = fs (r, h, hx, hy, ht) is the multiple density of a probability distribution
function. Therefore in common case the surface layer transformation can be written as follows (instead of eq. (2.1) or (3.10))
~ 1 r
u(zj h, t, r, h, hx, h , ht) = lim — J u(x, y, h, t)dxdydz
J SV®dVs SV
SV
where h = z1 / h( x, y, t )is fixed over the region of integration, h = z1 / h = const, SV is an arbitrary volume put in dV = LxLydz and containing dVs as a whole.
Here again we have the starting point of the theory of turbulence explained in second chapter. On this way we have lost the simplicity of transformation
(3.10), as there is an unknown dynamic roughness function h = h( x, y, t) in transformation (2.1).
The additive dynamic roughness surface model considered above is given by
h( x, y, t) = r ( x, y ) + hr ( x, y, t)
where hr (x, y, t )is the height of the viscous sublayer over the rough surface. Averaged this equation over a large area AxAy = LxLy we have: h = ra + hr, ht = hrt, where ra is the mean roughness height, i.e.
ra = J~T JJ r(x’ y )dxdy
x y AxAy
After replacing of the origin of the coordinate system in the new position z ® z - ra the dynamic roughness equation can be written as:
h1 (x, y,t) = r (x, y) - ra + hr (x, y,t) (3.11)
where h1= hr, h^t = hrt. Thus, we can imagine the smooth wall located at z = ra as it was defined by Schlichting [76] and the dynamic roughness surface with the dynamic roughness parameters given by (3.11). For this problem we should suggest that h1 > 0.
Note that the fluid flow near the plate surface z = ra is a typical heterogeneous flow included two parts: the roughness rigid elements part Sa = Sa (ra) and the fluid flow part equals to AS - Sa, where AS = LxLy. Put La = AS / Sa (ra) is the ratio of the whole area AS = LxLy to the roughness area Sa = Sa (ra) at z = ra. The roughness density parameter proposed by Dvorak [79] is given by Ls =AS / S, where S is the total roughness area. Since ra = akr / Ls, so La =La (ra) can be considered as a function of the Dvorak's roughness parameter: L a = L a (L s). For the roughness elements considered by Bettermann [77], Schlichting [76] and Coleman et. al. [83] this function can be calculated in the closure form.
For the roughness compounds by the spherical uniform elements
ra = 2kr / 3Ls , Sa (ra ) = S[1 - (1 - 2^ / k„ )2], hence
1 8 ( 2 Л 1 --
(3.12,a)
In the case of the surface roughened by spherical segments (Figure 3.1) we have: ra = kr (3 + kr 2 / r 2)/6L s, Sa (ra) = S(1 + rakr /r 2)(1 - ra / kr), therefore
1 Sa (ra) 1
La AS Ls
where, e = kr2 / r2.
In the case of the surface with conical uniform elements we have
Sa (ra ) = S(1 - ra / kf )2 , ra = kf /3Ls , thus
1 1 ' 1 '?
(3.12,b)
1
1 --
v 3L
(3.12,c)
In the case of two dimensional roughness as it has been considered by Bettermann [77], Dvorak [79] and Dalle Donne & Meyer [82] L a = L a (ra) depends only on the roughness elements width and pitch (see Figure 3.1):
La = L s = L / d (3.12,d)
The mean liquid surface between the roughness elements at z = ra equal to (1 -1/ La)AS, therefore the mean fluid density p = (1 -1/ La)p (note, that in the real case additionally some liquid volume can be excluded from the mean flow, hence it can be p = (1 - f / L a)p where f > 1 is the shape parameter counted for instance the liquid involved in the viscous sublayer around the roughness elements). The mean dynamic viscosity is defined as
Thus, L a = L a (ra) is the important parameter for the rough surface effects modelling because the boundary condition for the mean velocity gradient should be given at z = ra.
The mean velocity logarithmic profile in the turbulent flow over the rough surface can be derived from (2.27) written in the new coordinate system:
where ta is the effective shear stress applied to the effective smooth wall at z = ra. Thus for the dimensionless mean velocity gradient on the effective wall in common case one can propose that
As it follows from the mean velocity logarithmic profile in the turbulent flow established by Schlichting (see eq. (3.4)) the dimensionless turbulent length in the first equation (3.13) depends on the roughness parameters and thus can't be defined from an equation similar to eq. (2.24). To define 1+ note, that for the completely rough regime in the classical sense, when k+r >> 1, one can suppose that I»I0. Then the exact solution of the problem (3.13)-(3.14) can be written as
But this equation also follows from (38) if we put Rt = 0 in the non-linear model (24), and therefore € = I0 = 0. Hence, in the case of turbulent flow over a rough surface the main turbulent length scale can be defined as 1+ = 1/ kGa ^ 1+, and the second scale of the turbulent velocity equals zero. The mean velocity
m _ Pv _(1 -1/L a )m .
du+ _ exp(Д -1) $ _ h h Wdh
(3.13)
where z _ z - ra, I0 _ I(h ), h _ Z / h, щ _V/£ + hl .
mdu + /dz +_mGa _ mta /tw at z+_ 0 (3.14,b)
(3.15)
logarithmic profile follows from (3.15) at the long distance from the wall. Put z1 + >> 1/ kGa in (3.15), and then we have
+ 1 + -1 1
u =— ln z1 + c, with c = — ln(2kG ). k 1 k
This equation can be rewritten in the standard form as follows:
U + = kclnz1 + C0 - Klnkr -D(Ls) ’ (3.16)
D( L - > = C0 +
where c0 = 5.015. Note, that the finale result (3.16) mainly depends on the mean velocity gradient applied to the effective surface at z1+ = 0 .
There are two available cases which can be realised in the experimental situation: the roughness elements installed on the absolutely smooth surface and the roughness elements installed on the rough surface. In the first case we suppose that the mean velocity gradient applied to the effective smooth wall is proportional to the velocity gradient over a smooth surface given by the first equation (2.27) for z = ra. Used the boundary condition (3.14, b) we have:
(1 - f / L a )Ga _ br exp[ 10 - 1К )] (3.17)
k ra V1 + (Л0* / Г,* )2
where the shape parameter f > 1 introduced to estimate the frontal and leeward re-circulation zones effect, ra+ = rauT / v, br is the parameter. Suggested that
br ^1 + (1+/ra)2 = b0 where b0 is a function of the roughness parameter, we have (for the smooth background surface):
(1 - f/L,, )Ga = , g = exp[ 10 - I(r; )] (3.18)
kra
here g is the transitional layer parameter. Note, that for the high value of the roughness density parameter may be k+ >> 1 (completely rough regime in the classical sense), but simultaneously r^+ = ak+ / Ls < 1. Thus we have g = 1 for the completely rough regime that defined only for r^+ >> 1+ as it follows from the second equation (3.18). The main turbulent length scale can be estimated from (3.18) as 1+ = 1/kGa = r+ (1 - f /La)/b0g. Substituted Ga from (3.18) into the second equation (3.16) finally we have:
D( L ) + 11 a(1 - f / L a) ln g (3 19)
^ =C" + k'" 2b„ A, (3'19)
The rough surface effects model (3.19) depends on two parameters b0,f chosen from the best correlation with the experimental data. We should underline that b 0 is the friction parameter of the rough surface and f is the parameter of the mean density of fluid involved in the mean turbulent flow at the level
z = ra .
In the second case the mean velocity gradient model is the same as (3.19) but we should put ra = akr / L s + rg where the averaged height of the background
roughness is rg. Both models have been testified and shown the good agreement
with the experimental data.
3.3 Modelling of roughness density effect. 3D roughness elements
To test the roughness surface effect model (3.19) the turbulent flow data for 3D roughness elements obtained by Schlichting [76] and re-evaluated by Coleman et. al. [83] has been used. The main result reported by Coleman et. al. [83] is that some Schlichting's data was obtained probably in the transitionally rough regime. The experimental techniques in Schlichting's [76] and Coleman et. al. [83] experiments have been analyzed and it was surmised that Schlichting's data was measured in the fully rough regime but some details of his experimental technique have not been reported.
The computed (1) and experimental data by Schlichting (3) and Coleman et. al. (5) are shown in Figure 3.3 for spheres (Fig. 3.3,a), spherical segments (Fig.
3.3,b) and cones (Fig. 3.3,c). The points (4) are computed from the experimental data by Coleman et. al. [83] which has been corrected with transitional layer parameter g calculated on (2.22) as follows g = exp[1.264 - 0.805arctan(0.048ra+)].
As it shown in Figure 3.3 the transitional layer effect is essential for the plate with roughness in a form of spherical segments (two points with k+r = 14; 27 and consequently Ls = 31.8; 17.9) and conical elements (two points with k+r = 55; 211 and L s = 31.8; 17.9 respectively), and relatively small for all data with r^+ = ak+ / Ls > 16, including data for the plate roughened by spheres.
Note, that the experimental data re-evaluated by Coleman et. al. [83] is getting closer to the original Schlichting's data after the correction on the transitional layer effect. Therefore it seems to be clear that the data by Coleman et. al. [83] is rather based on another experimental technique then the original Schlichting's data [76].
Figure 3.3: Roughness density effect on the turbulent boundary layer for 3D roughness elements: a) spheres; b) spherical segments; c) cones; d) generalised correlation
The corrected data has been used to estimate the parameters b 0, f in the equation (3.19) which can be written for the completely roughness regime (g = 1) as follows
1 a(1 - f / Aa)
D(A,) = C0 + -1» 2b 0 As (3'20)
The roughness parameters a, Aa =Aa(ra) are given by (3.12, a-c) for the spheres, spherical segments and conical elements respectively.
As it has been established in the case of the plate with spheres b0 = 0.65, f = 1.11 for the data obtained by Coleman et. al. [83] (solid line (1) in Figure
3.3,a) and b0 = 0.4, f = 1.25 for the corrected points. For the roughness elements in the form of spherical segments b0 = 3, f = 1 (solid line (1), Figure 3.3,b) and for the conical elements b0 = 0.7, f = 1 - see Figure 3.3,c. For comparison the Bettermann-Dvorak's correlated line (2) also is shown in Figure 3.3.
The magnitude b0 can be explained in terms of the rough surface drag which has the same value for the spheres and conical elements and mach less for the surface with spherical segments. The mean fluid density parameter is f »1 for considered types of roughness elements. Note that in the case of the surface roughened by spheres the function D(As) has a maximum at As » 2.35 (as has
been established in numerical experiments the maximum location depends on the value f approximately as As » 2.175f2 3 for the range 1 < f < 2).
The experimental data for the surfaces with spheres, spherical segments or conical elements can be collected together used an "universal" parameter which is different from that proposed by Bettermann [77] Dvorak [79], Dirling [80], Simpson [81], Kind & Lawrysyn [87] and other. This correlation is available for the high roughness density parameter at A s >> 1, then
1 a
D(As) » c0 +-ln:
- 2b 0 A s
and therefore the "universal" parameter is given by A = b0As / a. The solid line (1) computed on the equations (3.20) for b0 /a = 1, f = 1.11 is shown in Figure 3.3, d with the corrected experimental data for the rough surfaces with spheres (2), spherical segments (3) and conical elements (4) . The classic sand-grain-roughened pipe flow experiment of Nikuradse (1933) with D =-3, As = 4/ p is presented by point (5). The hoar-frost roughness data of Kind & Lawrysyn [87] are plotted by points (6). Note, that data of Kind & Lawrysyn [87] has been corrected with transitional layer parameter
g = exp[1.264 - 0.805arctan(0.048ra+)]/ (1 + Asrg / akr)
where ra = akr / As + rg, a = 1/3, rg = krfr is the averaged height of the background roughness, fr depends on the frost formation and has been calculated for the plate 1-6 of Kind & Lawrysyn [87] as follows fr =.013;.04;0.12;0.12;.04;04. In this case we have b0 = 0 7 as for as for conical elements. The experimental data for 3D rounded elements of Simpson (1973) is shown by symbols (7), and for his data 0.45 < b0 / a < 0.55.
Thus one can suggest that the rough surface with spheres is the basic case for 3D roughness elements because all data shown in Figure 3.3,d is correlated well with the basic line (1).
Then one can propose the model for b considered this parameter as a function of the width-to-height ratio b 0= b 0(d / kr). For instance, at b J a = 1 we have the Dvorak's roughness density parameter A = As = DS / S. For a linear function b 0= b 0(d / kr) the "universal" parameter is related to that of Bettermann (1966) since in the case of transverse square bars S = d, a = 1 and hence A = b L / kr, where b i is the numerical value. Another forms of the roughness density parameter [80-81, 85-86] are based on an approximation of function b 0= b 0(d / kr), dependent on available experimental data and theoretical consideration of the roughness elements drag in turbulent flows.
3.4. Modelling of roughness density effect. 2D roughness elements
Using the roughness density parameter model in the form (3.12, d) and suggesting that g = 1 (completely roughness regime) one can write (3.19) for this case as follows
D(A •) = C0 + i'-f1 (3'21)
For the constant value of the parameters b 0, f the function D(As )has a maximum at A s = 2f. This maximum can be defined from the Dalle Donne & Meyer's model (3.6) as As = 6.3kr / d +1, and therefore f = (6.3kr / d +1)/2 . Thus as it follows from the experimental data the shape parameter varies with kr / d. To compare the experimental data with the arbitrary value of the shape parameter let us introduce the roughness density parameter in the form Af = As / f, then
the roughness surface effect model can be rewritten as
1 (1 -1/ A f) 1
D( A f) = £-0 + - In 2b1A * - - ln(df / kr) (3.22)
where b 1= b 0 kr / d.
In this model the experimental data for various kr / d can be plotted together as the graph of the function
Df (A f) = D( A f) +1/-ln(df / kr)
as well as in the Dalle Donne & Meyer's model (3.6). But as it has been established the shape parameter derived from the model (3.6) isn't a good approximation.
Note that in the common case one can suggest that f = 1 + f1 kr /d, where fkr is the total length of the frontal and leeward re-circulation zones. The model
(3.6) gives for this parameter an unreasoned result that f1 = 3.15-d/2kr. Thus the experimental data of various authors listed in Table 3.1 has been used to find the right form of f and b 0. The best correlation for about 130 points is given by
1 + , f = exp( - B4n B11 q1), A = b k-
(3.23)
where B = (1 + kr / d)/ Ls, q = 0.625, b = 0.12 .
Figure 3.4: The rough surface effect on the turbulent flow: 2D roughness elements data 1-24, the solid line is calculated on (3.22-3.23). The multiply fragment of correlation line is shown in the lower part
Figure 3.4 shows Df (L f) calculated on (3.22-3.23) - the solid line (1), and
the experimental data 1-24 of various authors listed in Table 3.1 (note, we have used values R(¥) from Table 2 of Dalle Donne & Meyer [82] instead of the original data 1-18). The symbols description is given in the right part of Figure
3.4 and in Table 3.1. A fragment of the correlated line is shown in the lower part of Figure 3.4. One can see that the predicted roughness density effect (solid line) is in a good agreement with the main experimental data.
Finally note, that model (3.22-3.23) is derived for the rough surface composed by the transverse rectangular rods and can't be applied to 2D roughness elements of another form without additionally verification.
3.5 Model of total length of frontal and leeward re-circulation zones
Analyzing expression (3.22) one can find two singular points: Lf ® 1 and Lf which correspond to two branches of function Df (Lf). Dalle Donne & Meyer [82] model (3.6) also has two singular points L*D ® 0 and L*D ®¥ . Taken into account that L*D = (d / kr)(Ls -1) one can conclude that these two singular points are located at Ls ® 1 and Ls ® ¥ accordingly. As we can see from the data shown in Figures 3.2 there is probably another singular point at L*d »1 . The data collected around the point at L*D »1 has been obtained mainly for kr / d = 1. Thus this point can be at Ls » 2 . But generally speaking what is the physical reason for this point? In Figure 3.5,a the normalised total length of the frontal and leeward re-circulation zones (solid lines) which depends on the Dvorak's roughness density parameter f = f1(Ls) and the mean fluid density (broken line) calculated for kr / d = 0.5; 2; 5 are shown.
As we can see from Figure 3.5,a the total length has a maximum located in a point L*s = L*s (kr / d). According to this the effective mean fluid density has a minimum which may be less then zero. As it follows from (3.14, b), if ta is limited value and m ® 0 then Ga ® ±¥ thus it is a singular point for the function D = D(L s). Physically it means that the frontal and leeward re-circulation zones have intersection. As it is well known in this case the skimming flow is realised. In the model (3.22)-(3.23) this regime is counted statistically and probably with some error. In any case the data over the point L*D »1 in Figure
3.2 is replaced to the point Lf = 1 in Figure 3.4. Note that correlated line goes throughout this data better in Figure 3.4 than in Figure 3.2.
An unexpected result has been found out in the numerical experiment that function D = D(Ls) has one maximum for kr / d < 1.436 and two maximum for kr /d > 1.436 as shown by the solid lines 1 in Figure 3.5,b calculated for kr /d = 5. This result is very sensitive to the variations of the value f . If f is multiplied by 0.9 then the function D = D( L s) louses the singular point and looks like solid line
2 in Figure 3.5,b. Now we have only experimental data shown in Figure 3.4 which is not sufficient to confirm this result.
A restriction for this model can be established if the length scale 1+ = r+ (1 - f /La)/b0g found out for the rough surface is compared with the main turbulent length scale 1+ computed for the boundary layer over a smooth surface as 1+ > 1+. It puts the limitation for the normalised mean fluid density as (1 - f / L a) > 1+ b 0 / r+ »1/ r+ for 2D roughness considered above. If this restriction is broken then it means that the model (3.22)-(3.23) also can't be used properly. Supposed that in this case 1+ = 1+ one can regularise the function D = D( L s) in the singular point shown in Figure 3.5,b.
3.6 About efficiency of air pollutants removal
The efficiency of the air pollutants removal from the urban streets depends on several parameters including the emission rate, wind speed and aerodynamic properties of the streets. For the one road system this problem is reduced to the estimation of aerodynamic properties of the street depending on the geometry of ambient buildings [16, 114]. In case of several parallel streets with uniform buildings, the problem mainly is similar to the task about turbulent flows over a rough surface with artificial 2D roughness elements which was considered above.
So, put kr is the scale of buildings height, L is the mean distance between streets, l is the typical width of roads; q is the specific emission of a typical vehicle, INi is the number of vehicles passing through chosen cross-section of a road on the street "i" per fixed time interval. Then, the specific emission on the
street 'i" is determined as q' = qN' /1. Note, that q is the technique characteristic of the vehicle.
^ - pip a) b)
FT Si ii CD D kr/d = 5
4-
■ /// 0-
'/ / 0.5 =
/ co i 1 -n-4—,— i 1—г
1 10 100 As 1 10 100 Л
Figure 3.5: a) the normalised total length of the frontal and leeward re-circulation zones (solid lines) versus the Dvorak's roughness density parameter f = f1(Ls) and the normalised mean fluid density (broken line) calculated for kr /d = 0.5; 2;5.; b) the roughness density effect on the mean velocity shift D = D(L s) at fixed kr / d = 5. The solid line 1 is calculated on (3.22)-(3.23), The solid line 2 is calculated also on (3.22)-(3.23), with parameter f decreased on 10%
Assuming that all emitted gases are removed from the street by turbulent flow, one can define the turbulent scale of impurity concentration on the street "i" as: C* = q' / u*. Hence, the impurity concentration near to the road is proportional to the emission rate and inversely proportional to the wind friction velocity. But the turbulent velocity scale in turn depends on a wind velocity on the external boundary of turbulent flow. Using the first equation (3.16) on the level z = H, one can written the wind velocity in the logarithmic layer
U 1 H -
— = — In—— u k k„
■ + c0
D
From here the wind friction velocity and turbulent scale of concentration can be defined as follows
1 H - z0
— ln---------------------+ c0 - D
k k„ 0
\-1
C * = q_
’ C U
1, H - z0
— ln—----------
k k„
+ c0 - D
(3.24)
These expressions can be used for an evaluation of efficiency of air pollutants removal from streets depending on the geometric parameters. Setting of the
wind velocity at height H = mkr >> z0, where m is simplifying the second expression (3.24), we have:
C * _ ч_
1 U
— lnm + c0 - D k
the numerical value and
(3.25)
In activities [41-42] and some other, it is offered to measure the wind velocity in urban environment at the level H = kr, therefore for m = 1, but it isn't right, because the mean velocity profile is not homogeneous on this height.
Thus, the problem about efficiency of air pollutants removal from the streets is reduced to that to minimise the expression (3.25). If the parameter of emission, wind velocity and the wind speed parameter m are given, then a minimum of the dynamic concentration is reached at a maximum of D(L s).
Figure 3.6: a) The shift of the mean velocity logarithmic profile versus parameter of roughness density; b) the maximum of mean velocity logarithmic profile shift vs the roughness elements height to width ratio
The function D = D( L s) computed for the various roughness elements height to width ratio kr /d = 0.25; 0.5; 1 is shown in Figure 3.6, a. Note that D(Ls) has only one maximum for the parameter kr / d from the interval 0 < kr / d < 1.436 , thus the problem of minimisation has only one solution. The numerical solution Dmax = Dmax(kr /d) is shown in Figure 3.6,b by the solid line 1. In the interval kr / d > 1.436 the function Dmax = D (kr / d) has two branches, but this solution is
r max max v r ' '
not shown. The position of maximum dependent on kr / d is shown in Figure 3.6, b by the solid line 2.
Finally note that only the case of the neutral stratified turbulent flow has been considered. The problem about turbulent diffusion in the atmospheric boundary layer with an arbitrary stratification over the region with several streets will be analysed in the subsection 5.2 of Part 5.
Therefore, the model of turbulent boundary layer over rough surface has been given and it has been stated that the profile of a mean velocity in turbulent boundary layer can be described by logarithmic function with the parameters, depending on geometry of elements of roughness. Model application in the case of three-dimension elements of roughness of the type of spheres, spherical segments and cones, as well as in the case of two-dimension elements of roughness of the type rectangular roods, placed on the flat surface with equal step has been considered. The basic parameters of model have been evaluated. Equations, describing summary volume of re-circulating zones and elements resistance to roughness have been obtained. The question about possibility of application the obtained results for modelling the efficiency of city streets ventilation have been considered.
(To be continued)
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