2D Velocity-Field Analysis Using Triple Decomposition of Motion

A straightforward application of the new triple decomposition of the local relative motion near a point to 2D velocity fields is presented. Unlike the Cauchy-Stokes double decomposition of motion into a pure straining motion and a rigid-body rotation the triple decomposition of motion aims, moreover, at the extraction of an effective pure shearing motion. It results in a more detailed flow description as the third term is responsible for a specific portion of vorticity labelled "shear vorticity" and for a specific portion of strain rate labelled "shear strain rate". Introduction The concept of the triple decomposition of motion has been motivated by a longstanding problem of a vortex and vorticity expressed by the fact that "solely vorticity cannot distinguish between swirling motions and shearing motions", Kida and Miura [1] (this fact is emphasized also by Jeong and Hussain [2] and Cucitore, Quadrio and Baron [3]). Let us add an analogous statement for strain rate: "solely strain rate cannot distinguish between straining motions and shearing motions". Vorticity and strain rate reflect the conventional Cauchy-Stokes decomposition of the local relative motion near a point into two elementary homogeneous motions: a pure irrotational straining motion along principal axes of the rate of strain tensor (generally including a uniform dilatation) and a rigid-body rotation. In Kolář [4] an arbitrary instantaneous state of the local relative motion near a point is decomposed into three elementary motions–each described by an additive part of u ∇ with a distinct tensor character–explicitly including an effective pure shearing motion. The first two elementary parts of the proposed triple decomposition remind, at least in tensor character, the two elementary parts of the conventional double decomposition, , and represent their residual portions after extracting an effective pure shearing motion. Ω S u + = ∇ Triple Decomposition of Motion The algorithm of the triple decomposition of motion is briefly summarized below. For details, justification and discussion see Kolář [4] where three elementary motions are introduced in terms of a virtually structured continuum. The triple decomposition of the local relative motion near a point aims, basically, at the extraction of an effective pure shearing motion. It reads (1) ( ) ( ) ( )SH RR EL u u u u ∇ + ∇ + ∇ = ∇ where the three elementary homogeneous motions are described by additive parts of : a pure irrotational straining motion given by symmetric tensor (subscript "EL" reminds the term "elongation"), a rigid-body rotation given by antisymmetric tensor , and an effective pure shearing motion u ∇ ( )EL u ∇ ( )RR u ∇ ( )SH u ∇ described below. A general pure shearing motion (that is, at this stage, without specification of the term "effective") is defined by a "purely asymmetric tensor form" of fulfilling in a suitable reference frame (null tensors are excluded, the subscript /j denotes differentiation) u ∇ (for all i, j). (2) 0 0 = = i j j i u OR u / / The desired reference frame, showing an effective pure shearing motion "in a clearly visible manner" described by the form (2), is called a basic reference frame (BRF). In this frame, the local relative motion is decomposed in terms of additive parts of u ∇ at a given point. In any other frames rotated with respect to the basic reference frame under an orthogonal transformation Q, an arbitrary elementary part of , say A, is described simply by QAQ u ∇ . Note that the sum of additive parts transforms as ∑ i i A . (3) ∑ ∑ = ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ i i i i T T Q QA Q A Q The triple decomposition then reads in the BRF (specified afterwards) BRF