Moment generating functions and scaling laws in the inertial layer of turbulent wall-bounded flows

Xiang Yang, Ivan Marusic, Charles Meneveau

    Research output: Contribution to journalArticle

    19 Citations (Scopus)

    Abstract

    Properties of single- and two-point moment generating functions (MGFs) are examined in the inertial region of wall-bounded flows. Empirical evidence for power-law scaling of the single-point MGF 〈exp(qu+)〉 (where u+ is the normalized streamwise velocity fluctuation and q a real parameter) with respect to the wall-normal distance is presented, based on hot-wire data from a Reτ = 13 000 boundary-layer experiment. The parameter q serves as a 'dial' to emphasize different parts of the signal such as high- and low-speed regions, for positive and negative values of q, respectively. Power-law scaling 〈exp(qu+)〉 ∼ (z/δ)-τ(q) can be related to the generalized logarithmic laws previously observed in higher-order moments, such as in 〈u+2p1/p, but provide additional information not available through traditional moments if considering q values away from the origin. For two-point MGFs, the scalings in 〈exp[qu+(x) + q′u+(x + r)]〉 with respect to z and streamwise displacement r in the logarithmic region are investigated. The special case q′ = -q is of particular interest, since this choice emphasizes rare events with high and low speeds at a distance r. Applying simple scaling arguments motivated by the attached eddy model, a 'scaling transition' is predicted to occur for q = qcr such that τ(qcr) + τ(-qcr) = 1, where τ(q) is the set of scaling exponents for single-point MGFs. This scaling transition is not visible to traditional central moments, but is indeed observed based on the experimental data, illustrating the capabilities of MGFs to provide new and statistically robust insights into turbulence structure and confirming essential ingredients of the attached eddy model.

    Original languageEnglish (US)
    Article numberR2
    JournalJournal of Fluid Mechanics
    Volume791
    DOIs
    StatePublished - Mar 1 2016

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    wall flow
    Wall flow
    Scaling laws
    scaling laws
    moments
    scaling
    low speed
    high speed
    vortices
    Boundary layers
    Turbulence
    dials
    Wire
    ingredients
    boundary layers
    turbulence
    exponents
    wire
    Experiments

    All Science Journal Classification (ASJC) codes

    • Condensed Matter Physics
    • Mechanics of Materials
    • Mechanical Engineering

    Cite this

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    title = "Moment generating functions and scaling laws in the inertial layer of turbulent wall-bounded flows",
    abstract = "Properties of single- and two-point moment generating functions (MGFs) are examined in the inertial region of wall-bounded flows. Empirical evidence for power-law scaling of the single-point MGF 〈exp(qu+)〉 (where u+ is the normalized streamwise velocity fluctuation and q a real parameter) with respect to the wall-normal distance is presented, based on hot-wire data from a Reτ = 13 000 boundary-layer experiment. The parameter q serves as a 'dial' to emphasize different parts of the signal such as high- and low-speed regions, for positive and negative values of q, respectively. Power-law scaling 〈exp(qu+)〉 ∼ (z/δ)-τ(q) can be related to the generalized logarithmic laws previously observed in higher-order moments, such as in 〈u+2p〉1/p, but provide additional information not available through traditional moments if considering q values away from the origin. For two-point MGFs, the scalings in 〈exp[qu+(x) + q′u+(x + r)]〉 with respect to z and streamwise displacement r in the logarithmic region are investigated. The special case q′ = -q is of particular interest, since this choice emphasizes rare events with high and low speeds at a distance r. Applying simple scaling arguments motivated by the attached eddy model, a 'scaling transition' is predicted to occur for q = qcr such that τ(qcr) + τ(-qcr) = 1, where τ(q) is the set of scaling exponents for single-point MGFs. This scaling transition is not visible to traditional central moments, but is indeed observed based on the experimental data, illustrating the capabilities of MGFs to provide new and statistically robust insights into turbulence structure and confirming essential ingredients of the attached eddy model.",
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    Moment generating functions and scaling laws in the inertial layer of turbulent wall-bounded flows. / Yang, Xiang; Marusic, Ivan; Meneveau, Charles.

    In: Journal of Fluid Mechanics, Vol. 791, R2, 01.03.2016.

    Research output: Contribution to journalArticle

    TY - JOUR

    T1 - Moment generating functions and scaling laws in the inertial layer of turbulent wall-bounded flows

    AU - Yang, Xiang

    AU - Marusic, Ivan

    AU - Meneveau, Charles

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    Y1 - 2016/3/1

    N2 - Properties of single- and two-point moment generating functions (MGFs) are examined in the inertial region of wall-bounded flows. Empirical evidence for power-law scaling of the single-point MGF 〈exp(qu+)〉 (where u+ is the normalized streamwise velocity fluctuation and q a real parameter) with respect to the wall-normal distance is presented, based on hot-wire data from a Reτ = 13 000 boundary-layer experiment. The parameter q serves as a 'dial' to emphasize different parts of the signal such as high- and low-speed regions, for positive and negative values of q, respectively. Power-law scaling 〈exp(qu+)〉 ∼ (z/δ)-τ(q) can be related to the generalized logarithmic laws previously observed in higher-order moments, such as in 〈u+2p〉1/p, but provide additional information not available through traditional moments if considering q values away from the origin. For two-point MGFs, the scalings in 〈exp[qu+(x) + q′u+(x + r)]〉 with respect to z and streamwise displacement r in the logarithmic region are investigated. The special case q′ = -q is of particular interest, since this choice emphasizes rare events with high and low speeds at a distance r. Applying simple scaling arguments motivated by the attached eddy model, a 'scaling transition' is predicted to occur for q = qcr such that τ(qcr) + τ(-qcr) = 1, where τ(q) is the set of scaling exponents for single-point MGFs. This scaling transition is not visible to traditional central moments, but is indeed observed based on the experimental data, illustrating the capabilities of MGFs to provide new and statistically robust insights into turbulence structure and confirming essential ingredients of the attached eddy model.

    AB - Properties of single- and two-point moment generating functions (MGFs) are examined in the inertial region of wall-bounded flows. Empirical evidence for power-law scaling of the single-point MGF 〈exp(qu+)〉 (where u+ is the normalized streamwise velocity fluctuation and q a real parameter) with respect to the wall-normal distance is presented, based on hot-wire data from a Reτ = 13 000 boundary-layer experiment. The parameter q serves as a 'dial' to emphasize different parts of the signal such as high- and low-speed regions, for positive and negative values of q, respectively. Power-law scaling 〈exp(qu+)〉 ∼ (z/δ)-τ(q) can be related to the generalized logarithmic laws previously observed in higher-order moments, such as in 〈u+2p〉1/p, but provide additional information not available through traditional moments if considering q values away from the origin. For two-point MGFs, the scalings in 〈exp[qu+(x) + q′u+(x + r)]〉 with respect to z and streamwise displacement r in the logarithmic region are investigated. The special case q′ = -q is of particular interest, since this choice emphasizes rare events with high and low speeds at a distance r. Applying simple scaling arguments motivated by the attached eddy model, a 'scaling transition' is predicted to occur for q = qcr such that τ(qcr) + τ(-qcr) = 1, where τ(q) is the set of scaling exponents for single-point MGFs. This scaling transition is not visible to traditional central moments, but is indeed observed based on the experimental data, illustrating the capabilities of MGFs to provide new and statistically robust insights into turbulence structure and confirming essential ingredients of the attached eddy model.

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