12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

SCALE-SPACE ENERGY DENSITY AND INHOMOGENEOUS EFFECT IN TURBULENTCHANNEL FLOW

Fujihiro Hamba
Institute of Industrial Science
The University of Tokyo
Komaba, Meguro-ku Tokyo 153-8505, Japan
hamba@iis.u-tokyo.ac.jp

ABSTRACTA new expression for the scale-space energy density basedon filtered velocities was proposed to clarify the reason for thenegative value of the energy density and to better understandand predict inhomogeneous turbulent flows. The newexpression consists of homogeneous and inhomogeneousterms; the former is always non-negative whereas the latter canbe negative because of the turbulence inhomogeneity. DNS
data of turbulent channel flow was used to examine the twoterms of the turbulent energy and energy density. It was shownthat a concave profile of the turbulent energy near the wallaccounts for the negative value of the energy density very close
to the wall.

∞K = ∫ dkE(k)0(5)E(k)[= 2πk Q(k)] ≥ 0The spectrum E(k) can be considered an energy density in thewavenumber space and its integral is equal to the kinetic
energy K. Moreover, E(k) is non-negative because of (3). We
can obtain a better understanding of turbulence by examining
the energy transfer in the wavenumber space.
For inhomogeneous turbulence, the Fourier transformcannot be always used because of complex flow domain andboundaries. Instead of the energy spectrum in the wavenumber
space, the second-order structure function 〈δu 2i (x,r)〉[ δu i (x,r) = u i (x + r) − u i (x) ] in the physical space can beused to describe the kinetic energy at the length scale of r. The
transport equation for the structure function has been examined
for several inhomogeneous turbulence (Cimarelli et al., 2013;
Cimarelli et al., 2016; Gatti et al., 2020). However, thestructure function represents the kinetic energy of eddies withsizes equal to or less than r; it is not the energy density in thescale space. It is more appropriate to introduce the energy
density that satisfies (Davidson 2004)

INTRODUCTION
For homogeneous isotropic turbulence, the energy
spectrum is commonly used to describe the scale dependenceof turbulent fluctuations. The energy transfer has been studied
in detail, and several closure theories have been developed. In
contrast, one-point statistical quantities are mainly employed
for inhomogeneous turbulence, such as the turbulent kineticenergy. The scale dependence of the energy must be importantfor understanding and predicting inhomogeneous turbulent
flows.
The velocity field is decomposed into mean and fluctuatingparts.
(1)u *i (x) = U i (x) + u i (x), U i = 〈u *i 〉

∞

∞

∞

∞

−∞−∞(6)E(x,r) ≥ 0(7)where the turbulent energy K at a position is decomposed into
the non-negative energy density E(x,r) . Using the two-point
velocity correlation, the author proposed an expression for theenergy density in the scale space (Hamba, 2015; Hamba, 2018).
The energy density always satisfies (6), but satisfies (7) only
for homogeneous turbulence. It can be negative forinhomogeneous turbulence.
In this work, we improve an expression for the energy
density using three filtered velocities (Hamba 2022). The newenergy density consists of homogeneous and inhomogeneousparts. The expression can still be negative for inhomogeneous
turbulence, but the reason for the negative value can beidentified in relation to the turbulence inhomogeneity. We
analyze DNS data of turbulent channel flow to examine thehomogeneous and inhomogeneous parts of the energy density
and investigate the reason for its negative value.

u i (x) = ∫ dku i (k)exp(ik ⋅ x), ∫ dk = ∫ dk x ∫ dk y ∫ dk z (2)〈u 2i 〉 = ∫ dkQ ii (k), 〈u i (k)u †i (k)〉 = Q ii (k)δ(0)

K(x) = ∫ drE(x,r)
0

For homogeneous turbulence, the Fourier transform is used todescribe the scale dependence of the velocity fluctuation asfollows:
−∞

(4)

2(3)

The velocity fluctuation u i (x) is decomposed into Fouriermodes in the wavenumber space in (2). Moreover, the velocity
variance 〈u 2i 〉 is also decomposed in the wavenumber space in(3). As a result, the turbulent kinetic energy K(= 〈u 2i 〉 / 2) isexpressed in terms of the energy spectrum as follows:1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
homogeneous and inhomogeneous parts. For homogeneousturbulence, only the homogeneous part remains non-zero; it is! (x,s) . For
clearly non-negative because it is the variance Qii
inhomogeneous turbulence, the energy density can be negative
because of the inhomogeneity of the energy Q ii (x,s) . We canquantitatively examine the deviation from the non-negative partin the case of inhomogeneous turbulence.

ENERGY DENSITY IN SCALE SPACEFor simplicity, we describe the formulation in the case ofone-dimensional filtering. The three-dimensional filtering isdescribed in Hamba (2022). The first filtered velocity is anordinary one and defined as∞u i (x,s) = ∫ dx ′G(x − x ′,s)u i ( x ′ )−∞G(x − x ′,s) =⎡ (x − x ′ )2 ⎤1exp ⎢ −⎥1/22s ⎦(2πs)⎣(8)
ANALYSIS OF CHANNEL FLOW
To investigate the homogeneous and inhomogeneous partsof the energy density, we examine the DNS data of a turbulent
channel flow. The size of the computational domain is
L x × L y × L z = 2π × 2 × π , where x, y, and z denote thestreamwise, wall-normal, and spanwise directions, respectively.
Physical quantities are nondimensionalized by the friction
velocity u τ and channel half width L y / 2 . The Reynolds
number is set to Re τ = 590 .
In this paper, we treat the one-dimensional filtering in thewall-normal direction. The first filtered velocity is written as(9)Here, we adopt a quantity s with a dimension of the square ofthe length and refer to s as the scale. This filtered velocity
represents the velocity with a scale equal to or greater than s.
By differentiating u i (x,s) with respect to s, we can obtaina filtered velocity with a scale equal to s. The second filteredvelocity is defined asû i (x,s) ≡ −∞∂u i (x,s) = ∫ dx ′Ĝ(x − x ′,s)u i ( x ′ )−∞∂s(10)

1u i (x, y,z,s) = ∫ dy′G(y, y′,s)u i (x, y′,z)⎡ (x − x ′ )2 ⎤1 ⎡ 1 (x − x ′ )2 ⎤Ĝ(x − x ′,s) =−⎥ exp ⎢ −⎥ (11)
1/2 ⎢22s ⎦(2πs) ⎣ 2s2s⎦⎣−1

The filter function G(y, y′,s) is basically the same as (9), but itshould be modified near the wall. We define G(y, y′,s) as thesolution of the following equation

The first filtered velocity and the original velocity can bewritten as∞∞s

0u i (x,s) = ∫ ds′û i (x, s′ ), u i (x) = ∫ dsû i (x,s)∂1 ∂2G(y, y′,s) =G(y, y′,s), G(y, y′,0) = δ(y − y′ ) (21)∂s
2 ∂y2with the boundary condition at the wall at y = ±1 . Considering
the solenoidal condition for u i (x,s) , we adopt the following
two sets of boundary conditions:(12)

The velocity can be decomposed in terms of û i (x,s) in thescale space. Moreover, the following relationship is obtained.
∂1 ∂2
1 ∂2u(x,s) =u (x,s), û(x,s) = −u (x,s) (13)
2 i
∂s
2 ∂x
2 ∂x 2 i
By differentiating u i (x,s) with respect to x, we introduce
another filtered velocity defined as∞∂! − x ′,s)u ( x ′ )u! i (x,s) ≡u (x,s) = ∫ dx ′G(xi−∞∂x i(14)⎡ (x − x ′ )2 ⎤1x − x′exp ⎢ −⎥1/2s2s ⎦(2πs)⎣(15)! − x ′,s)) = −G(xwhich satisfies∞K(x) = ∫ ds 12 Q̂(x,s)
0i∂G= 0 for u x and u z , G = 0 for u y∂y(23)The energy density given by (19) is written as11!
1 ∂2(24)
Q̂ ii (y,s) = Q(y,s)−Q (y,s)22 ii
4 ∂y2 ii
By integrating each term with respect to s, we can obtain an
expression for the turbulent energy as∞
∞
1!
1 ∂2
1K(y) = ∫ ds Q(y,s) − ∫ ds
Q (y,s) + Q ii (y,∞)
ii
00
2
4 ∂y2 ii2(25)where the right-hand side consists of the homogeneous,inhomogeneous, and residual parts.
First, we plot instantaneous profiles of the filtered velocity
to show the effect of the filtering given by (8). Figure 1 shows
one-dimensional profiles of the velocity u x (x, y,z) and thefiltered velocity u x (x, y,z) as functions of y at x = z = 0 in thecase of the second boundary condition for G(y, y′,s) . The
scale s is set to 0.0016 for Fig. 1(b) and 0.016 for Fig. 1(c). The
scale s = 0.0016 corresponds to a filter width s1/2 = 0.04
normalized by the channel half width. It is clearly seen that thevelocity profile become smoother as the scale increases.
Hereafter, we examine profiles of averaged quantities.
Figure 2 shows the profiles of the turbulent energy and its threeparts given by (25) as functions of y in the case of the firstboundary condition for G(y, y′,s) . Except for the near-wallregion, the inhomogeneous part is small and the homogeneouspart is dominant in Fig. 2(a). However, in Fig. 2(b) where thenear-wall region is shown, we can see that the inhomogeneous(17)(18)

Using the variance of the third filtered velocity,! (x,s) = 〈u! (x,s)u! (x,s)〉 , and the relationship given by (13),Qii(22)

andrepresents the kinetic energy of eddies with scales equal to or
greater than s. By differentiating Q ii (x,s) with respect to s, we
can define the energy density Q̂ ii (x,s) / 2 , where∂Q (x,s) = 2〈û i (x,s)u i (x,s)〉∂s ii∂G= 0 for u y∂y

G = 0 for u x and u z ,The third filtered velocity u! i (x,s) is the first derivative ofu i (x,s) whereas the second filtered velocity û i (x,s) is thesecond derivative of u i (x,s) .
Using the three filtered velocities, we propose a newexpression for the energy density in the scale space. The
variance of the first filtered velocity written as(16)
Q ii (x,s) = 〈u i (x,s)u i (x,s)〉Q̂ ii (x,s) ≡ −

(20)

i

we can rewrite the energy density as follows
11!
1 ∂2(19)Q̂ ii (x,s) = Q(x,s) −Q (x,s)ii
2
24 ∂x 2 ii
A similar relationship was also used to investigate thestretching of vorticity by the strain rate in Johnson (2020). We
call the first and second terms on the right-hand side of (19) the212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
part shows a large negative value. The magnitude of both parts
is greater than 10 and seems too large compared to the peakvalue of K (= 4.6) .
Figure 3 shows the profiles of the turbulent energy and its
three parts in the case of the second boundary condition for
G(y, y′,s) . The inhomogeneous part shows a negative value
near the wall, but its magnitude is much smaller than that in Fig.
2. The turbulent kinetic energy is expressed mainly in terms of
its homogeneous part and is slightly modified by its
inhomogeneous part near the wall. In this sense, the second setof boundary condition given by (23) is more appropriate.
Hereafter, we will show the results given by this boundarycondition. The inhomogeneous part involved in (24) is minus
the second derivative of the velocity variance. This implies thatthe negative value of the inhomogeneous part at y+ < 5 shownin Fig. 3(b) is caused by the concave profile of the turbulentenergy near the wall.
Figure 4 shows the contours of the pre-multiplied! (y,s) / 2 , in thehomogeneous part of the energy density, sQii−4s − y plane. The peak is located at s = 6.5 × 10 and+y = −0.94 ( y = 35 ). The peak location with respect to s
increases as y increases. The contour plots are very similar to
those of ry E(y,ry ) given by Hamba (2018). Figure 5 shows the! (y,s) / 2 as functions of s at four y locations.
profiles of sQii
The shift of the peak towards large scales is clearly shown. Thevalue in the case of y+ = 30 is the largest and this factcorresponds to the peak location of the homogeneous partshown in Fig. 3(b).
Figure 6 shows the profiles of the pre-multiplied energy
density and its two parts given by (24) as functions of s at twoy locations. In Fig. 6(a) for y+ = 86 , the homogeneous part ispositive whereas the inhomogeneous part shows both positive
and negative values. Because the inhomogeneous part is notvery large, the total value is always positive. In contrast, in Fig.
6(b) for y+ = 2.8 , the inhomogeneous part has a large negativevalue at s = 10−4 , and the resulting total value also shows anegative value. As discussed before, the large negative value ofthe inhomogeneous part at y+ = 2.8 is caused by the concave
profile of the turbulent energy near the wall. Therefore, theenergy density can be negative in the region very close to thewall because of the inhomogeneous effect.

REFERENCES
Cimarelli, A., De Angelis, E., and Casciola C. M., 2013,"Paths of energy in turbulent channel flows", J. Fluid Mech.,Vol. 715, pp. 436-451.
Cimarelli, A., De Angelis, E., Jimenez, J., and Casciola C.
M., 2016, "Cascades and wall-normal fluxes in turbulent
channel flows", J. Fluid Mech., Vol. 796, pp. 417-436.
Davidson, P. A., 2004, Turbulence, Oxford University
Press.
Gatti, D., Chiarini, A., Cimarelli, A., and Quadrio M., 2020,"Structure function tensor equations in inhomogeneous
turbulence", J. Fluid Mech., Vol. 898, A5 1-33.
Hamba, F., 2015, "Turbulent energy density and its
transport equation in scale space", Phys. Fluids, Vol. 27, p.
085108.
Hamba, F., 2018, "Turbulent energy density in scale spacefor inhomogeneous turbulence", J. Fluid Mech., Vol. 842, pp.
532-553.
Hamba, F., 2022, "Scale-space energy density forinhomogeneous turbulence based on filtered velocities", J.
Fluid Mech., Vol. 931, A34 1-24.
Johnson, P. L., 2020, "Energy transfer from large to small
scales in turbulence by multiscale nonlinear strain and vorticity
interactions", Phys. Rev. Lett., Vol. 124, 104501 1-6.

CONCLUSIONS
A new expression for the energy density in the scale space
was proposed using three filtered velocities. The new
expression consists of homogeneous and inhomogeneous parts.
The homogeneous part is always non-negative whereas theinhomogeneous part is proportional to the second derivative of
the velocity variance. DNS data of a turbulent channel flow
were used to evaluate the homogeneous and inhomogeneousparts of the energy density and the turbulent energy. It was
shown that a concave profile of the filtered-velocity variance at
the wall accounts for the negative value of the energy density
in the region very close to the wall. In future work, we will
examine the energy transfer between different scales using theproposed energy density. It would also be interesting toformulate a statistical theory using filtered velocities toimprove the turbulence models.

3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

5(a)

Total
Homogeneous
Inhomogeneous
ResidualTurbulent energy

43210-1-2-1.0
Figure 1. Profiles of velocity u x (x, y,z) and filtered velocityu x (x, y,z) as a function of y at x = z = 0 in the case of thesecond boundary condition for G(y, y′,s) (a) u x (b) u x fors = 0.0016 (c) u x for s = 0.016

Turbulent energy

100-2
04050

Figure 3. Profiles of turbulent energy K and its three parts
given by (25) in the case of the second boundary condition for
G(y, y′,s) (a) −1 ≤ y ≤ 1 (b) 0 ≤ y+ ≤ 50 .

1.0

10-0.25-0.4

y0.0
50.0-0.6

0.10-10102030

400.1

0.050.2-0.8Total
Homogeneous
Inhomogeneous
Residual-5-15030+

0.150.5200.10.0y10yTotal
Homogeneous
Inhomogeneous
Residual(b) 15Turbulent energy

Total
Homogeneous
Inhomogeneous
Residual

2-1-0.5

1.05Turbulent energy

35-15-1.00.5

410-100.0y5(b)(a) 15-5-0.5-1.010-610-510-410-3

s10-2

10-1

100101Figure 4. Contour plots of pre-multiplied homogeneous part! (y,s) / 2 .
sQ50+yFigure 2. Profiles of turbulent energy K and its three parts
given by (25) in the case of the first boundary condition for
G(y, y′,s) (a) −1 ≤ y ≤ 1 (b) 0 ≤ y+ ≤ 50ii

Energy density

10010-1

10-210-310-410-510+

y=-0.995 y =2.8+
y=-0.982 y =11+
y=-0.950 y =30+
y=-0.854 y =86+
y=-0.702 y =176+
y=-0.214 y =464-7-5-3-1

1101010s
Figure 5. Profiles of pre-multiplied homogeneous part! (y,s) / 2 as functions of s.
sQii

41012th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022(a) 0.5
Energy density

0.4

Total
Homogeneous
Inhomogeneous

0.3
0.2
0.1
0.0-0.1-71010-5-310s10-1

101(b) 0.4
Energy density

0.2
0.0-0.2-0.4

Total
Homogeneous
Inhomogeneous

-0.6
-0.8
-7
1010-5-3-1

1101010s
Figure 6. Profiles of pre-multiplied energy densitysQ̂ ii (y,s) / 2 and its two parts given by (24) (a) for y+ = 86
and (b) for y+ = 2.8 .

5