Flow Visualisation

Initial attempts at coherent structure visualisation commonly focussed on simple flow parameters such as pressure minima or vorticity magnitude. Using pressure minima can cause errors from either minima which do not involve vortical motion or the elimination of vorticity based minima from viscous effects. Using|ω|, however, fails in shear flows as the shear contributes to the vorticity measured.

The eigenvalues (σ) of the velocity gradient tensor ∇u satisfies the charac- teristic equation:

σ3−P σ2+Qσ−R= 0,

whereP =ui,i, Q= 1₂(u2_i,i−ui,juj,i) andR = det(ui,j) are the three invariants of

∇u. Hunt et al. [1988] used a positive second invariantQ to define a vortex core. Chong et al. [1990] noted that if ∇u has complex eigenvalues at a spatial location then the the velocity distribution is either closed or spiralled around that point, moving with its reference frame. This occurs when:

∆ = 1 3Q 3 + 1 2R 2 >0.

Let S and Ω denote the symmetric and antisymmetric components of ∇u, respectively, such that:

Si,j =

1

2(ui,j+uj,i) and Ωi,j = 1

2(ui,j−uj,i). Using these definitionsQcan be written as1₂ ||Ω||2_{− ||S||}2

, where||A||2_{= tr(AA}T_).

Physically, the values of ||S||2 _{and ||Ω||}2 _{correspond to the shear strain rate and} vorticity magnitude, respectively. From this notation the Q > 0 criterion can be considered as defining a vortex core as a region in which the vorticity magnitude is greater than the shear strain rate.

Nk>1, where Nk is defined [Truesdell, 1953] by:

Nk=

||Ω|| ||S||

Note that although this limit ofNk>1 is equivalent to theQ >0 condition, theNk

value is independent of the strength of the vorticity, and can therefore be considered as a measure of the quality of the rotation.

The λ2 criterion presented by Jeong and Hussain [1995] is derived from the use of pressure minima to visualise vortices. By calculating a matrix equation for the pressure hessian with the unsteady straining and viscous effects removed, a more accurate representation of a vortex core is derived. It is shown that the existence of two negative eigenvalues of the matrix S2 + Ω2 constitutes a vortex core. If

λ1 ≥λ2 ≥λ3 are the eigenvalues ofS2+ Ω2 then two negative eigenvalues occur if

λ2 is negative.

Following the development of a vortex identification scheme in Jeong and Hussain [1995], Jeong et al. [1997] went on to study coherent structures in uncon- trolled channel flow. The profile of λ2 was calculated, where the average is taken over time and the x−z plane. This showed that there is a balance in the λ2 val- ues between the region inside and outside of the vortex cores. A strong correlation between−λ2 and |ωx|in the region 10< y+<40 highlights the strong streamwise

alignment of near-wall structures. An average near-wall structure was educed for both structures with positive and negative vorticity. It is found that the structures each have an inclination of 9◦ and a tilting angle of 4◦ in opposite directions. The spatial relationship between the two structures is found via correlation and ensemble averaging of ωx. The conditionally averaged velocity fields are also presented and

2.8

3D Turbulent Boundary Layers

A turbulent boundary layer is considered to be three-dimensional if the mean flow direction varies with the distance from the wall. There are many previous studies of such type, and can either be driven by a spanwise pressure gradient or moving wall in order to create the spanwise shear. In the case of wall motion a stationary or opposing wall is often present to ensure spanwise shear. The spanwise wall forcing techniques which are the focus of the current work are similar, but have a periodicity and constantly varying wall velocity.

Experimental works have been performed using various methods to create the spanwise shear. Anderson and Eaton [1989] hypothesised that the shear caused by the spanwise pressure difference would affect the different vorticity directions in different ways. Asymmetry was found in the conditional averaging around ejection and sweep events by Littel and Eaton [1994]. This suggested that strong sweep events are associated with rotation opposed to the spanwise shear and strong ejec- tions are related to rotation enhanced by the spanwise wall motion. Kang et al. [1998] performed a similar averaging, separating contributions to the local velocity by quadrant analysis. This indicated that the asymmetry was related to the events which produce negative Reynolds shear stress, and that the events which caused positiveu0_v0 _{were symmetric.}

Coleman et al. [1996] performed three variations of simulation to understand the effect of using different methods to create the near-wall shear. Two of which use wall motion, either abruptly moving or stopping the wall. The third uses a transverse strain setting ∂U_∂x = −∂W_∂z . The greatest drag reduction was seen when the shear was applied in the region 5< y+ <15. Le et al. [2000] used an impulsive spanwise moving wall with velocityW+

s =−8.5. The drag is seen to initially decrease before

shear stress,γτ, angles. These are defined as: γs= arctan ∂W/∂y ∂U/∂y (2.7) γτ = arctan v0w0 u0v0 (2.8)

The lag angle, quantified byλ=γs−γτ, is initially large but reduces greatly as the

streamwise shear recovers. This compares with calculations in previous experiments [Littel and Eaton, 1994]. Conditional averaging around sweeps and ejections showed that ejections were stronger than sweeps whenω_x0+was positive, whereas sweeps were stronger than ejections withω_x0 negative. This was attributed to the near-wall tilting angle of the coherent structures, caused by the spanwise shear.

Jung and Sung [2006] simulated the flow through a concentric annulus via DNS, in which the inner pipe is rotating perpendicular to the mean flow direction. Using the coherent structure eduction method [Jeong et al., 1997] an average λ2 structure was found for both positive and negativeω_x0+. The negative vortex, which opposed the wall motion, was lifted up, away from the wall, and therefore had larger inclination angle of 13◦, compared to the slightly decreased value of 8◦ for the positive vortex. The rotation of the inner pipe affected the upstream ends of the structures causing an angling of both in the same direction. The tilting angle was stronger in the positive vortex at −20◦, whereas the negative vortex had an angle of−8◦. Studying the field of Reynolds shear stress around the averaged structures shows that the crossflow reduces strong sweeps from the negative vortex and strong ejections from the positive one.

Studying a three-dimensional flow caused by adverse pressure gradient on a Couette flow, Holstad et al. [2010] also looked at equilibrium effects. This work focused on channel flow to remove any implications of the centrifugal force. The angles studied in Coleman et al. [1996] were plotted, as well as the mean velocity angle and the intensity angle. It was seen that the intensity angle is larger than

the mean velocity angle, whereas the turbulent shear stress angle is smaller. The intensity and turbulent shear stress angles are similar toward the channel centre and vary almost linearly. The mean velocity and mean velocity gradient angles are similar in the near wall region, but separate as the velocity gradient tends to zero near the upper wall. Studying the vorticity showed that ωx is increased in

the buffer region from the 2D flow. This was attributed to the mean shear in the spanwise directions which implies an increase in the fluctuations in this direction. For the same reason the spanwise vorticity fluctuation is reduced, and the wall- normal fluctuation unchanged.