Flow through a cylinder bundle

(a) Geometry of a flow between two cylinders along with the interrogation window where time-resolved PIV measurements are taken. This simple configuration inherits the main features from more complex settings, including the presence of multiple, distinct frequencies and the ‘‘flip-flop oscillations’’ of the exit stream. (b) A snapshot of velocity vectors and vorticity contours, resulting from time-resolved PIV measurements, illustrates the oscillatory motion of the exiting jet. Data courtesy of Electricite de France (EdF).

Flow through cylinders in a bundle configuration is often encountered in the utility and energy conversion industry. Owing to their versatility and efficiency, cross-flow heat exchangers account for the vast majority of heat exchangers in oil refining, process engineering, petroleum extraction, and power generation sectors. Consequently, even a modest improvement in their effectiveness and operational margins would have a significant impact on production efficiency. In spite of their widespread use, the details of the flow (and heat transfer) through cylinder bundles are far from fully understood, thereby making this type of flow, along with its simplified variants, the subject of active research.

Vortex shedding and complex wake interactions can induce vibrations and structural resonances which, in turn, may result in fretting wear, collisional damage, material fatigue, creep, and ultimately in cracking. Even though the flow through a cylinder bundle is highly complicated, it is characterized by well-defined shedding frequencies. For single-row cylinders, only a few distinct frequencies are detected, while for more complex array configurations, a multitude of precise shedding frequencies are observed. The presence of distinct shedding frequencies makes this type of flow well-suited for a decomposition of the flow fields into single-frequency modes.

In addition to the multiple-frequency behavior, a characteristic oscillatory pattern – labelled as a ‘‘flip-flop phenomenon’’ – is typically observed in the exit stream of the flow through cylinder bundles. It consists of a meta-stable deflection of the exiting jet off the centerline location. A statistical analysis of the velocity field in a cross-stream section shows a strongly bimodal distribution. Over a wide range of Reynolds numbers, the Strouhal number associated with the oscillation between these two states is approximately Reynolds number independent. This flow feature can also be represented by a dynamic mode decomposition of the flow fields.

Snapshots of velocity vectors and vorticity contours resulting from a sparse DMD representation (with modes) of the full data sequence. A deflected jet and strong vortical components off the symmetry axis are observed. An animation of this flow field is shown below.

Dependence of the absolute value of the DMD amplitudes on (a) the frequency and (b) the real part of the corresponding DMD eigenvalues .

Performance loss, , of the optimal vector of amplitudes resulting from the sparsity-promoting DMD algorithm with DMD modes.

Eigenvalues resulting from the standard DMD algorithm (circles) along with the subset of eigenvalues selected by its sparsity-promoting variant (crosses). The dashed curves identify the unit circle.

Dependence of the absolute value of the amplitudes on the frequency (imaginary part) of the corresponding eigenvalues . The results are obtained using the standard DMD algorithm (circles) and the sparsity-promoting DMD algorithm (crosses) with DMD modes.

Animation

An animation of the flow field resulting from a sparse DMD representation (with modes) of the full data sequence is shown next. We observe a deflected jet and strong vortical components off the symmetry axis. This animation has been obtained using the frequency identified by the sparsity-promoting DMD algorithm and it reproduces the main features of the full data sequence, i.e., the lateral swaying of the jet under the influence of the wake vortices of the two cylinders.