Abstract
The nonlinear decay of a disturbance is calculated by an iterative solution to the Navier-Stokes equations. The general expressions obtained for successive approximations are suitable for machine computations, within the limitations of the computer capacity. For the initial condition a three-dimensional cosine distribution with two harmonic terms is assumed. The nonlinear interaction of these harmonic terms then produces new harmonics. As the iteration process proceeds, a large number of harmonics, at wave numbers higher or lower than, or the same as the original ones, are generated. This process appears to be similar to that occurring in the early stages of the development of turbulence by flow through a grid.
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Abbreviations
- a, b :
-
dimensionless Fourier coefficients for pressure fluctuation, x 20 a*/ρν 2, x 20 b*/ρν 2 (see (8))
- a*, b*:
-
Fourier coefficients for pressure fluctuation
- a i, bi, ci, di :
-
dimensionless Fourier coefficients for velocity component, x 0 a * i /ν, x 0 b * i /ν, etc. (see (6) and (12))
- a * i , b * i :
-
Fourier coefficients for velocity component
- a 0 i , b 0 i :
-
initial dimensionless Fourier coefficients for velocity components, x 0 a 0* i /ν, x 0 b 0* i /ν (see (4))
- a 0* i , b 0* i :
-
initial Fourier coefficients for velocity components
- a c i, κ, q , a s i, κ, r , a c i, n, q , a c i, m, r , a s i, n, q , a s i, m, r :
-
constants given by equations (24), (25), (28), (29), (32), and (33)
- A c i, κ , A s i, κ , A c i, n , A c i, m , A s i, n , A s i, m :
-
dimensionless Fourier coefficients for velocity components, x 0 A c * i, κ /ν
- A c * i, κ , A s * i, κ :
-
Fourier coefficients for velocity component
- b c κ, q , b s κ, r , b c n, q , b c m, r :
-
constants given by equations (24), (25), (28), (29), (32), and (33)
- m i , n i :
-
components of dimensionless wave number vectors, x 0 m * i , x 0 n * i ; particular values of κ i
- m * i , n * i :
-
components of wave number vectors
- m, n :
-
dimensionless wave number vectors
- m, n :
-
magnitudes of dimensionless wave number vectors
- p :
-
dimensionless pressure x 20 p*/ρν 2
- p*:
-
pressure
- q i , r i :
-
components of initial dimensionless wave number vectors x 0 q * i , x 0 r * i
- q, r :
-
initial dimensionless wave number vectors
- q * i , r * i :
-
components of initial wave number vectors
- t :
-
dimensionless time, νt*/x 20
- t*:
-
time
- u i :
-
component of dimensionless velocity, x 0 u * i /ν
- u * i :
-
component of velocity
- u c,c i,n,m , u Emphasis> s,s i,n,m , u Emphasis> s,c i,n,m :
-
interaction terms given by equations (30), (34), and (36)
- x i :
-
dimensionless position coordinate, x * i /x 0
- x * i :
-
position coordinate
- x :
-
dimensionless position vector
- x 0 :
-
characteristic length
- κ :
-
dimensionless wave number vector; x 0 κ*
- κ*:
-
wave number vector
- ν :
-
kinematic viscosity
- ρ :
-
density
- *:
-
on certain dimensional quantities
- s:
-
on quantities in sine term
- c:
-
on quantities in cosine terms
- —:
-
over averaged quantities
References
Taylor, G. I. and A. E. Green, Proc. Roy. Soc. (London) A 158 (1937) 499.
Jain, P. C., Numerical Study of the Navier-Stokes Equations for the Production of Small Eddies from Large Ones, Un. of Wisconsin MRC-TSR-491, Madison (Wisc.) 1964.
Burgers, J. M., Advances of Applied Mechanics, vol. 1, p. 171, edited by R. von Mises and T. von Kármán, Academic Press, New York 1948.
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Deissler, R.G. Nonlinear decay of a disturbance in an unbounded viscous fluid. Appl. sci. Res. 21, 393–410 (1969). https://doi.org/10.1007/BF00411623
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DOI: https://doi.org/10.1007/BF00411623