Flow normal to a disk-like plate

The flow normal to a stationary disk-like plate is investigated by direct numerical simulations across wide ranges of Reynolds number

(

100 less than or equals upper R e less than or equals 300 100 ≤ R e ≤ 300 $100 \leq Re \leq 300$

), number of polygon vertices

upper N N $N$

(

2 less than or equals upper N less than or equals 5 2 ≤ N ≤ 5 $2 \leq N \leq 5$

), and compression ratio

k k $k$

(

0 less than or equals k less than or equals 1 0 ≤ k ≤ 1 $0 \leq k \leq 1$

), where

k equals 0 k = 0 $k = 0$

and 1 correspond to a regular polygon and circular disk, respectively. All plates share identical area

upper A Subscript r A r $A_r$

and diameter–thickness aspect ratio

chi equals upper D divided by t Subscript d Baseline equals 50 χ = D / t d = 50 $\chi = D/t_d=50$

, with the nominal characteristic diameter defined as

upper D equals StartRoot 4 upper A Subscript r Baseline divided by pi EndRoot D = 4 A r / π $D = \sqrt {4A_r/\pi }$

. Five flow regimes are observed for all disk-like plates, denoted as: (i) steady geometric-symmetric, (ii) steady reflectional-symmetric, (iii) reflectional symmetry breaking, (iv) reflectional symmetry-preserving (RSP), and (v) chaotic. The wake transition depends strongly on the number of vertices

upper N N $N$

, particularly for

upper N less than or equals 3 N ≤ 3 $N \leq 3$

. For an elliptical plate (

upper N equals 2 N = 2 $N = 2$

), the flow exhibits strong symmetry until it reaches the chaotic state. At

k equals 0.4 k = 0.4 $k=0.4$

, an unusual RSP state is observed, characterised by periodic vortex shedding and a stable drag coefficient. Conversely, for triangular plates (

upper N equals 3 N = 3 $N = 3$

), the wake possesses only weak symmetry, and once symmetry is broken, it never recovers; the RSP state is absent for all

k k $k$

studies. For quadrilateral (

upper N equals 4 N = 4 $N = 4$

) and pentagonal (

upper N equals 5 N = 5 $N = 5$

) plates, the wake transition sequence encompasses all five flow regimes, closely resembling that of a circular disk, and providing a reference for polygonal plates of higher vertex count. The drag force exhibits a distinct parity effect: it oscillates at twice the shedding frequency for even-

upper N N $N$

plates, but contains both the fundamental and harmonic frequencies for odd-

upper N N $N$

plates. The compression ratio

k k $k$

modulates the wake symmetry and transition. When the plate becomes nearly circular (

k equals 0.8 k = 0.8 $k=0.8$

), the wake symmetry plane can deviate from the geometric symmetry plane of the plate. As

k k $k$

increases, the geometry converges towards a circular disk, weakening the edge vortices and lowering the critical Reynolds numbers towards the disk values.