### Citations

885 |
A First Course in Turbulence
- Tennekes, Lumley
- 1972
(Show Context)
Citation Context ...treamwise similarity coordinates. is performed over several iterations during which the eigenvalues are tracked using the following energy normalization: ǫ(n)p = Λ (n) p − Λ (n) p,o Λ (n) p,o × 100%, =-=(16)-=- where Λ(n)p,o = λ (n) p∑ n λ (n) p . (17) The resulting operation is shown in figure 10a for the first three POD modes under a range of grid densities between 0.01 < ∆ξ/Ll < 1.5. As ∆ξ/Ll → ∞ the ene... |

312 | The proper orthogonal decomposition in the analysis of turbulent flows
- Berkooz, Holmes, et al.
- 1993
(Show Context)
Citation Context ...ric form of the technique involves a linear integral equation of the Fredholm type following the Hilbert-Schmidt theory for symmetric integral kernels: ∫ Rij(x, x ′)φ (n) j (x ′)dx′ = λ(n)φ (n) i (x) =-=(1)-=- Here Rij(x, x ′) is a time averaged cross-correlation tensor constructed from an ensemble of multiple, statistically stationary measurements. The solution to (1) produces an ordered sequence of eigen... |

45 | Model reduction for compressible flows using pod and galerkin projection
- Rowley, Colonius, et al.
- 2004
(Show Context)
Citation Context ...ving the following integral eigenvalue problem: ∫ Rpp(ξ, ξ ′)φ(n)p (ξ ′)dξ′ = λ(n)p φ (n) p (ξ), (14) using a kernel constructed in streamwise similarity coordinates: Rpp(ξ, ξ ′) = 〈p(ξ, t)p(ξ′, t)〉. =-=(15)-=- It is the author’s understanding that the written form of the integral eigenvalue problem using streamwise similarity coordinates was first proposed by W. K. George.5 In figure 9, the kernel’s topolo... |

26 |
Reconstruction of the Global Velocity Field in the Axisymmetric Mixing Layer Utilizing the Proper Orthogonal Decomposition.
- Citriniti, George
- 2000
(Show Context)
Citation Context ...ctions can be used to reconstruct the original instantaneous velocity, ui(x, t) = ∑ n a(n)(t)φ (n) i (x), (2) using random and uncorrelated expansion coefficients, a(n)(t) = ∫ ui(x, t)φ (n)∗ i (x)dx. =-=(3)-=- whose mean square energies are the eigenvalues themselves: λ(n) = 〈a(n)a(q)〉δ(n,q). Typically the convergence of the POD eigenfunctions are shown to display how the energy containing events are ranke... |

22 |
R.F.,“Turbulence in the noise-producing region of a circular jet.”,
- Bradshaw, Ferriss, et al.
- 1964
(Show Context)
Citation Context ... exist, they are limited to the resolution of the measured grid. The finite number of eigenfunctions can be used to reconstruct the original instantaneous velocity, ui(x, t) = ∑ n a(n)(t)φ (n) i (x), =-=(2)-=- using random and uncorrelated expansion coefficients, a(n)(t) = ∫ ui(x, t)φ (n)∗ i (x)dx. (3) whose mean square energies are the eigenvalues themselves: λ(n) = 〈a(n)a(q)〉δ(n,q). Typically the converg... |

9 |
A hierarchy of lowdimensional models for the transient and post-transient cylinder wake.
- BR, Afanasiev, et al.
- 2003
(Show Context)
Citation Context ...physical coordinate on account of the growth in the turbulent motions as they evolve downstream. The two-point correlation coefficient is given by, ρpp(ξ, ξ ′) = 〈p(ξ, t)p(ξ′, t)〉 [p(ξ)2p(ξ′)2] 1 2 , =-=(13)-=- where ξ = ln(x − xo) and xo is the location of a virtual origin: determined from observation to be around x/Ds = −0.45. A good collapse of the two-point correlation coefficient in figure 8a shows how... |

8 |
Application of Multi- point Measurements for Flow Characterization
- Glauser, George
- 1992
(Show Context)
Citation Context ...ϑ)dϑ. (6) The eigenvalues produced from the solution to (5) are normalize by the total resolved energy at each streamwise station in the flow, Λ (n) i,o (x) = ∑ m λ (n) i (x;m)∑ n ∑ m λ (n) i (x;m) . =-=(7)-=- 3 of 12 American Institute of Aeronautics and Astronautics Paper 2009-0068 (a) (b) (c) −0.1 0 0.1 0.2 0.3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 η(x) U /U j x/D=3.0 x/D=4.0 x/D=5.0 x/D=6.0 x/D=7.0 x... |

8 |
The near pressure field of co-axial subsonic jets
- Tinney, Jordan
- 2008
(Show Context)
Citation Context ...rmed over several iterations during which the eigenvalues are tracked using the following energy normalization: ǫ(n)p = Λ (n) p − Λ (n) p,o Λ (n) p,o × 100%, (16) where Λ(n)p,o = λ (n) p∑ n λ (n) p . =-=(17)-=- The resulting operation is shown in figure 10a for the first three POD modes under a range of grid densities between 0.01 < ∆ξ/Ll < 1.5. As ∆ξ/Ll → ∞ the energy in the first POD eigenvalue increases ... |

7 |
Orthogonal decomposition of the axisymmetric jet mixing layer including azimuthal dependence.
- Glauser, George
- 1987
(Show Context)
Citation Context ... kernel is constructed from a Fourier-azimuthal transformation of the ensemble averaged two-point velocity correlation: Bij(r, r ′, x;m) = 1 2π ∫ pi −pi 〈ui(r, x, θ, t)uj(r ′, x, θ + ϑ, t)〉e−i(mϑ)dϑ. =-=(6)-=- The eigenvalues produced from the solution to (5) are normalize by the total resolved energy at each streamwise station in the flow, Λ (n) i,o (x) = ∑ m λ (n) i (x;m)∑ n ∑ m λ (n) i (x;m) . (7) 3 of ... |

6 |
Downstream evolution of the most energetic modes in a turbulent axisymmetric jet at high Reynolds number. Part 2. The far-field region.
- Gamard, Jung, et al.
- 2004
(Show Context)
Citation Context ...olutions and the solutions obtained using the under resolved grids, the following energy normalization was calculated for each POD mode, ǫ (n) i (x) = Λ (n) i (x)− Λ (n) i,o (x) Λ (n) i,o (x) × 100%. =-=(9)-=- 4 of 12 American Institute of Aeronautics and Astronautics Paper 2009-0068 (a) (b) 3 4 5 6 7 80.2 0.3 0.4 0.5 0.6 x/D L l /D u v 3 4 5 6 7 8 0.2 0.4 0.6 0.8 1 1.2 1.4 x/D r/D u v (c) 3 4 5 6 7 8 1.5 ... |

5 |
Coherent structure in a turbulent jet via a vector implementation of the proper orthogonal decomposition,”
- Iqbal, Thomas
- 2007
(Show Context)
Citation Context ...f these PIV measurements, this analysis is confined to integral lengthscales along the transverse coordinate of the jet (r), defined here as, Ll,i(r, x) = ∫ r′ r Rii(r, r ′, x) σi(r, x)σi(r′, x) dr′. =-=(8)-=- These lengthscales are computed for both streamwise and radial components of velocity. In figure 2 (a), the transverse integral lengthscales are extracted from two radial locations in the jet shear l... |

5 | Characteristic eddy decomposition of turbulence in a channel
- Moin, Moser
- 1989
(Show Context)
Citation Context ...Taylor microscale (T ), calculated here using the following series of definitions: Ll(x) = ∫ X Rpp(x, x ′) σ(x)σ(x′) dx′, (10) Lτ (x) = ∫ ∞ 0 |ρp(x, τ)|dτ, (11) T (x) = [2〈p(x, t)2〉 〈pt(x, t)2〉 ] 1 2 =-=(12)-=- where Rpp(x, x ′) = 〈p(x, t)p(x′, t)〉, ρp(x, τ) = 〈p(x, t)p(x, t ′)〉 and pt is the time derivative of the pressure: (dp/dt). The definition for the Taylor microscale is taken from 6.4.8 of Tennekes &... |

3 |
Twopoint similarity in the round jet
- Ewing, Frohnapfel, et al.
(Show Context)
Citation Context ...ng events are ranked and is calculated 2 of 12 American Institute of Aeronautics and Astronautics Paper 2009-0068 from the following energy normalization: Λ (n) i,o (x) = λ (n) i (x)∑ n λ (n) i (x) . =-=(4)-=- The denominator in (4) is simply the total cumulative energy and is equal to the total resolved turbulent kinetic energy: κi(x). Since we are ultimately dealing with a discretized system, the integra... |

2 | Low-dimensional characteristics of a transonic jet. Part 1. Proper orthogonal decomposition. - 30TINNEY, GLAUSER, et al. - 2008 |

1 |
Workshop on Jet Aeroacoustics. Summer 2006
- 5George
(Show Context)
Citation Context ... from a number of well documented applications of the POD technique to axisymmetric shear flows6, 3, 9, 8 and is written here as:∫ Bij(r, r ′, x;m)φ (n) j (r ′, x;m)r′dr′ = λ(n)(x;m)φ (n) i (r, x;m), =-=(5)-=- where the kernel is constructed from a Fourier-azimuthal transformation of the ensemble averaged two-point velocity correlation: Bij(r, r ′, x;m) = 1 2π ∫ pi −pi 〈ui(r, x, θ, t)uj(r ′, x, θ + ϑ, t)〉e... |

1 |
The structure of inhomogenous turbulent flows
- 10Lumley
- 1967
(Show Context)
Citation Context ...es with the integral length scale (Ll), the integral time scale (Lτ ) and the Taylor microscale (T ), calculated here using the following series of definitions: Ll(x) = ∫ X Rpp(x, x ′) σ(x)σ(x′) dx′, =-=(10)-=- Lτ (x) = ∫ ∞ 0 |ρp(x, τ)|dτ, (11) T (x) = [2〈p(x, t)2〉 〈pt(x, t)2〉 ] 1 2 (12) where Rpp(x, x ′) = 〈p(x, t)p(x′, t)〉, ρp(x, τ) = 〈p(x, t)p(x, t ′)〉 and pt is the time derivative of the pressure: (dp/d... |

1 |
Stochastic Tools in Turbulence
- 11Lumley
- 1970
(Show Context)
Citation Context ...(Ll), the integral time scale (Lτ ) and the Taylor microscale (T ), calculated here using the following series of definitions: Ll(x) = ∫ X Rpp(x, x ′) σ(x)σ(x′) dx′, (10) Lτ (x) = ∫ ∞ 0 |ρp(x, τ)|dτ, =-=(11)-=- T (x) = [2〈p(x, t)2〉 〈pt(x, t)2〉 ] 1 2 (12) where Rpp(x, x ′) = 〈p(x, t)p(x′, t)〉, ρp(x, τ) = 〈p(x, t)p(x, t ′)〉 and pt is the time derivative of the pressure: (dp/dt). The definition for the Taylor ... |

1 |
On low-dimensional Galerkin models for fluid flow
- 14Rempfer
- 2000
(Show Context)
Citation Context ...tegral eigenvalue problem For this component of the analysis, the sensitivity study is performed by solving the following integral eigenvalue problem: ∫ Rpp(ξ, ξ ′)φ(n)p (ξ ′)dξ′ = λ(n)p φ (n) p (ξ), =-=(14)-=- using a kernel constructed in streamwise similarity coordinates: Rpp(ξ, ξ ′) = 〈p(ξ, t)p(ξ′, t)〉. (15) It is the author’s understanding that the written form of the integral eigenvalue problem using ... |