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A generalized von Karman interpolation formula
Volume 133, number 3
PHYSICS LETTERS A
7 November 1988
A GENERALIZED VON KARMAN INTERPOLATION FORMULA R.A. CARHART Department of Physics, University of Illinois at Chicago, Chicago, IL 60680, USA
and A.B. KOSTINSKI Department ofElectrical Engineeringand Computer Science, University ofIllinois at Chicago, Chicago, IL 60680, USA Received 26 July 1988; accepted for publication 30 August 1988 Communicated by R.C. Davidson
A parametrized family of three-dimensional angle-averaged spectral functions is proposed that well approximates existing formulas for the energy-containing portion of the spectrum (low-k) in free shear turbulence. The family includes the von Karman interpolation formula as a special case, and is based on it. Varying the parameter allows the height and width of the spectral peak to be varied smoothly, while maintaining the smooth interpolation between origin and inertial subrange behavior. The properties of this family of spectra are presented in a convenient non-dimensionalized form. As an example of the usefulness of the family of spectra, it is applied to the centerline turbulent energy of an axisymmetric jet in the far field to illustrate the effect of low-k spectral shape on the shift of the peak with Reynolds number (Re) when the energy is constant with Re.
The three-dimensional (3D) energy spectral function has been well-specified and studied in the inertial subrange and, to a lesser extent, in the dissipation subrange. Its low-k structure (the energycontaining scales) is less understood, and has not been reliably estimated on a physical basis. Von Karman [1] advanced an interpolation formula that matches smoothly the origin behavior of the spectral function and the inertial subrange portion. That formula allows only one of many possible low-wavenumber spectral shapes. In this note we propose a generalization of the von Karman formula into a parametrized family of formulas with varying peak heights and widths. It approximates well all currently used spectral functions. The 3D spectral function, E(k), is defined by the following relationships:
q=Qu2=~(u’2+z/2+w’2)= JdkE(k), 0
(1)
E(k) ~=
=
a~213F( k) exp [ a ( k~)2], —
(2)
Jdkk2E(k)=a2/3$F(k)exp[_a(k~)2J. (3)
The overbar represents the average of a component, while the prime indicates the fluctuating component. The three components refer to an orthogonal coordinate system, and ü, the component along the mean flow direction, is used to non-dimensionalize the energy. The spectral function F(k) must equal k513 in the inertial subrange, and must be proportional to k4 near k=O. As usual, ~ represents the energy dissipation. The value of a in the spectrum may vary somewhat from one free shear flow to another, but it has some claim to at least approximate universality. The Saffman formula is used to approximate the dissipation spectrum, with the value ofa determined from eq. (3). The Kolmogorov scale ~ is defined in the usual way as (v3/)”4. Several seemingly unrelated forms have been used 149
Volume 133, number 3
PHYSICS LETTERS A
for F(k) in spectral studies. One commonly used spectral assumption is the Tennekes.-.Lumley formula [2], which has some theoretical basis as the solution of a highly approximated differential equation: (4) F(k)=k513 exp[— l.25(k/km)413]. This formula scales in the indicated way in terms of its peak, km, with the coefficient 5/4 in the exponential. Another commonly used formula, which we call the “primitive spectrum”, also scales in this way: F(k)=k~5/3(k/km)4
— —
k ~ 5/3 (k/km
[5/12+ (k/km)2]
‘6
17/6
“
‘
Each of these spectra has a different “shape” in the energy-containing range and scales with the location of the spectral maximum, km, as shown. But the primary difference between the spectra to us seems to lie in the height and width of the peak. This has led us to generalize the von Karman formula in a way that is simple, represents appropriate behaviors for all three size scales, and contains a single parameterthat adjusts the height and width of the peak smoothly: k~513(k/k
F(n, k)= r5/l2+ ‘k/k
)4
(7)
)‘hl 17/3n m
Let Yk/km. In terms of y, each of the spectral functions in eqs. (4)—(7) is just k~513times a dimensionless function of y. The generalized von Karman (GVK) formula (7) becomes the von Karman spectrum (6) for n = 2, and for n—+ it becomes the primitive spectrum (5). For n= 1.55 it differs from the Tennekes—Lumley formula (4) by at most 2.5% down to y= 0.65. (The Tennekes—Lumley formula has the wrong origin behavior, and the differential equation from which it was obtained is certainly invalid below the peak.) The height and widths of the peak (to half-height both above and below the peak) are shown in fig. 1 as continuous functions of n. The height of the peak varies from 0.14 for n = 1 to 1.00 for n = The width to half-height below the ~,
~.
150
~m~ax ~.
I~TT
-
(5)
k
Finally, there is the long-used suggestion of von Karman [1]: F’ k
7 November 1988
10°
10’ Exponent(n)
102
Fig. I. Variation of the height, upper half-width (~y~) and lower half-width (~\y_)with the exponent n in eq. (7) asa function of Re.
peak ranges from 0.64 to 0.16, decreasing as n increases; and the upper half-width ranges from 2.3 to 0.52. The sum of these two half-widths (the equivalent ofthe full width at half maximum) ranges from nearly 3.0 down to about 0.7. The von Karman spectrum (n =2) and the Tennekes—Lumley spectrum (n= 1.55) have comparable heights and widths. The von Karman spectrum is about 33% higher, 18% narrower above y= 1 and 9% narrower below y= 1 than the Tennekes—Lumley spectrum. We propose this formula as a simple representation of the width and height of the energy-containing portion of the spectrum, as well as one that gives the correct inertial subrange and dissipation subrange (with either the Saffman factor or some more sophisticated dissipation formulation). As more becomes known about the low-k behavior of the spectral function, it will be possible to specify a value for the exponent n. Since the sharpness of this peak affects the lengths of the inertial subrange and the origin values of the measurable one-dimensional (1 D) spectral functions, even presently accessible experiments can put limits on the value of n. This family of formulas can be useful, for example, in sub-grid modeling studies to verify that the large-scale structures are not modified by the residual effect of the low-k spectral shape implicit in the
Volume 133, number 3
PHYSICS LETTERS A
[12] C.A. Friehe, C.W. Van Atta and C.H. Gibson, in: Turbulent shear flows, AGARD Conf. Proc. No. 93 (1972) p. 1 [Re= 120000 and measurement of]. [13] F.H. Champagne, J. Fluid Mech. 80 (1978) 67 [Re=370000]. [14] M.M. Gibson, Fluid Mech. 15(1963)161 [Re=500000].
7 November 1988
[15] C. Friehe, private communication (1987) [Re= 1000000]. [16] R.A. Antonia, B.R. Satyaprakash and A.K.M.F. Hussain, Phys. Fluids 23 (1980) 695. [17] A. Kostinski and R.A. Carhart, Phys. Lett. A 107 (1985) 120.
153
PHYSICS LETTERS A
7 November 1988
A GENERALIZED VON KARMAN INTERPOLATION FORMULA R.A. CARHART Department of Physics, University of Illinois at Chicago, Chicago, IL 60680, USA
and A.B. KOSTINSKI Department ofElectrical Engineeringand Computer Science, University ofIllinois at Chicago, Chicago, IL 60680, USA Received 26 July 1988; accepted for publication 30 August 1988 Communicated by R.C. Davidson
A parametrized family of three-dimensional angle-averaged spectral functions is proposed that well approximates existing formulas for the energy-containing portion of the spectrum (low-k) in free shear turbulence. The family includes the von Karman interpolation formula as a special case, and is based on it. Varying the parameter allows the height and width of the spectral peak to be varied smoothly, while maintaining the smooth interpolation between origin and inertial subrange behavior. The properties of this family of spectra are presented in a convenient non-dimensionalized form. As an example of the usefulness of the family of spectra, it is applied to the centerline turbulent energy of an axisymmetric jet in the far field to illustrate the effect of low-k spectral shape on the shift of the peak with Reynolds number (Re) when the energy is constant with Re.
The three-dimensional (3D) energy spectral function has been well-specified and studied in the inertial subrange and, to a lesser extent, in the dissipation subrange. Its low-k structure (the energycontaining scales) is less understood, and has not been reliably estimated on a physical basis. Von Karman [1] advanced an interpolation formula that matches smoothly the origin behavior of the spectral function and the inertial subrange portion. That formula allows only one of many possible low-wavenumber spectral shapes. In this note we propose a generalization of the von Karman formula into a parametrized family of formulas with varying peak heights and widths. It approximates well all currently used spectral functions. The 3D spectral function, E(k), is defined by the following relationships:
q=Qu2=~(u’2+z/2+w’2)= JdkE(k), 0
(1)
E(k) ~=
=
a~213F( k) exp [ a ( k~)2], —
(2)
Jdkk2E(k)=a2/3$F(k)exp[_a(k~)2J. (3)
The overbar represents the average of a component, while the prime indicates the fluctuating component. The three components refer to an orthogonal coordinate system, and ü, the component along the mean flow direction, is used to non-dimensionalize the energy. The spectral function F(k) must equal k513 in the inertial subrange, and must be proportional to k4 near k=O. As usual, ~ represents the energy dissipation. The value of a in the spectrum may vary somewhat from one free shear flow to another, but it has some claim to at least approximate universality. The Saffman formula is used to approximate the dissipation spectrum, with the value ofa determined from eq. (3). The Kolmogorov scale ~ is defined in the usual way as (v3/)”4. Several seemingly unrelated forms have been used 149
Volume 133, number 3
PHYSICS LETTERS A
for F(k) in spectral studies. One commonly used spectral assumption is the Tennekes.-.Lumley formula [2], which has some theoretical basis as the solution of a highly approximated differential equation: (4) F(k)=k513 exp[— l.25(k/km)413]. This formula scales in the indicated way in terms of its peak, km, with the coefficient 5/4 in the exponential. Another commonly used formula, which we call the “primitive spectrum”, also scales in this way: F(k)=k~5/3(k/km)4
— —
k ~ 5/3 (k/km
[5/12+ (k/km)2]
‘6
17/6
“
‘
Each of these spectra has a different “shape” in the energy-containing range and scales with the location of the spectral maximum, km, as shown. But the primary difference between the spectra to us seems to lie in the height and width of the peak. This has led us to generalize the von Karman formula in a way that is simple, represents appropriate behaviors for all three size scales, and contains a single parameterthat adjusts the height and width of the peak smoothly: k~513(k/k
F(n, k)= r5/l2+ ‘k/k
)4
(7)
)‘hl 17/3n m
Let Yk/km. In terms of y, each of the spectral functions in eqs. (4)—(7) is just k~513times a dimensionless function of y. The generalized von Karman (GVK) formula (7) becomes the von Karman spectrum (6) for n = 2, and for n—+ it becomes the primitive spectrum (5). For n= 1.55 it differs from the Tennekes—Lumley formula (4) by at most 2.5% down to y= 0.65. (The Tennekes—Lumley formula has the wrong origin behavior, and the differential equation from which it was obtained is certainly invalid below the peak.) The height and widths of the peak (to half-height both above and below the peak) are shown in fig. 1 as continuous functions of n. The height of the peak varies from 0.14 for n = 1 to 1.00 for n = The width to half-height below the ~,
~.
150
~m~ax ~.
I~TT
-
(5)
k
Finally, there is the long-used suggestion of von Karman [1]: F’ k
7 November 1988
10°
10’ Exponent(n)
102
Fig. I. Variation of the height, upper half-width (~y~) and lower half-width (~\y_)with the exponent n in eq. (7) asa function of Re.
peak ranges from 0.64 to 0.16, decreasing as n increases; and the upper half-width ranges from 2.3 to 0.52. The sum of these two half-widths (the equivalent ofthe full width at half maximum) ranges from nearly 3.0 down to about 0.7. The von Karman spectrum (n =2) and the Tennekes—Lumley spectrum (n= 1.55) have comparable heights and widths. The von Karman spectrum is about 33% higher, 18% narrower above y= 1 and 9% narrower below y= 1 than the Tennekes—Lumley spectrum. We propose this formula as a simple representation of the width and height of the energy-containing portion of the spectrum, as well as one that gives the correct inertial subrange and dissipation subrange (with either the Saffman factor or some more sophisticated dissipation formulation). As more becomes known about the low-k behavior of the spectral function, it will be possible to specify a value for the exponent n. Since the sharpness of this peak affects the lengths of the inertial subrange and the origin values of the measurable one-dimensional (1 D) spectral functions, even presently accessible experiments can put limits on the value of n. This family of formulas can be useful, for example, in sub-grid modeling studies to verify that the large-scale structures are not modified by the residual effect of the low-k spectral shape implicit in the
Volume 133, number 3
PHYSICS LETTERS A
[12] C.A. Friehe, C.W. Van Atta and C.H. Gibson, in: Turbulent shear flows, AGARD Conf. Proc. No. 93 (1972) p. 1 [Re= 120000 and measurement of]. [13] F.H. Champagne, J. Fluid Mech. 80 (1978) 67 [Re=370000]. [14] M.M. Gibson, Fluid Mech. 15(1963)161 [Re=500000].
7 November 1988
[15] C. Friehe, private communication (1987) [Re= 1000000]. [16] R.A. Antonia, B.R. Satyaprakash and A.K.M.F. Hussain, Phys. Fluids 23 (1980) 695. [17] A. Kostinski and R.A. Carhart, Phys. Lett. A 107 (1985) 120.
153