Developing Flow in a Curved Rectangular Duct

Experiments by Kim and Patel

Description

• Developing turbulent flow in a 90 deg. curved duct of rectangular cross-section.
• Duct with two straight and one curved sections.
• Rectangular cross-section of the duct with a width of H = 20.3 cm and a height of 6H.
• Inner radius of curvature of the bend: R_i = 3×H.
• Straight upstream section with a length of 7.5H = 1.52 m.
• Straight downstream section with a length of 25.5H = 5.18 m.

Flow Characteristics

The initially 2D boundary layers developing on the vertical lateral walls are subjected to strong streamwise curvatures and associated pressure gradients along the bend. On the other hand, the pressure-driven secondary motion in the corner regions eventually leads to the formation of a longitudinal vortex on the convex wall. The duct aspect ratio is such that these two features of the flow develop more or less independently, without interaction.

Flow Parameters

• Air with a kinematic viscosity: =1.45 × 10^-5 m²/s.
• Freestream velocity at station U1 (x = -4.5H): U_o = 16 m/s.
• Reynolds number: U_oH/ = 224,000.

Inflow Conditions

At station U1 (x = -4.5H), the velocity is uniform in the core flow, outside the boundary layers, within a deviation less than 1%. On the vertical lateral walls, the boundary layers are of flat-plate type with a momentum thickness Reynolds number of 1650, a boundary layer thickness of = 0.08H and a friction coefficient of C_f = 0.0038. The 2D wind-tunnel contraction located 3H upstream of U1 introduces a secondary motion in the boundary layers on top and bottom flat walls but its magnitude reaches only 5% of the freestream velocity. The following measurements are then provided for a slightly three-dimensional duct flow but are sufficiently detailed to be used as inlet conditions.

• Velocity vectors: (files mu1@@@.dat) V/U_o, W/U_o
• Contour maps of: (files mu1@@@.dat)
  • First order moment U/U_o
  • Second order moments
    
    • Reynolds stresses: /U_o², /U_o², /U_o², /U_o², /U_o²
    • Turbulent kinetic energy: k/U_o²(deduced)
• Wall friction coefficient,C_f, distribution on each wall (files su1in.dat, su1out.dat and su1up.dat).

Experimental Details

Hot-wire velocity measurements have been carried out using a miniature X-wire probe for the turbulence quantities.

Mean velocity measurements have been carried out using a five-hole pressure probe of a diameter of 3 mm.

All velocity measurements have been made in the upper half of the duct divided into 5 different domains, namely in1, up1, ou1, in2, ou2 (see file readme).

Wall stress measurements using two pressure probes in combination (only the magnitude is actually measured). The friction coefficient is defined as C_f = 2/(U_o²).

Static pressure measurements using wall taps. The pressure coefficient is defined as C_p = 2(p - p_o)/(U_o²), where p_o is the static pressure at (0,0,3H).

Measurement Errors

(U)	1.5%
(V), (W)	3%
()	5%
(other Reynolds stresses)	10%

Available Measurements

The following measurements are available at 1 station upstream of the bend: U2 (x=-0.5H); 3 stations along it: 15, 45, 75 and 2 stations downstream of it: D1 (x=0.5H), D2 (x=4.5H).

• Velocity measurements ($$ is the station and @@@ the domain name)
  • Velocity vectors: V/U_o, W/U_o
  • Contour maps of: (files m$$@@@.dat)
    • First order moment U/U_o
    • Second order moments
      • Reynolds stresses: /U_o², /U_o², /U_o², /U_o², /U_o²
      • Turbulent kinetic energy: k/U_o²(deduced)
    • Wall friction coefficient, C_f, distribution on each wall (files s$in.dat, sout.dat and s$up.dat).
• Pressure measurements (files pressure.dat and pressure.tab)

The following measurements are available along the inner and the outer walls, in the plane of symmetry.

• Distribution of the static pressure coefficient C_p

Instructions for calculations

Inlet conditions:

The calculation of the duct flow should be started at station U1 using the experimental values provided as inlet conditions. The non-measured quantity may be assumed as negligible.

Symmetry:

Due to geometric symmetry with respect to the z=0 plane, one can use a computational domain including only the upper half of the duct.

Outlet conditions:

The outlet should be placed sufficiently far away (x > 30H) so that zero gradients may be assumed for the flow variables.

Presentation of results:

The following results should be plotted and compared with the data:

At all locations:

1. mean secondary velocity vectors
2. mean streamwise velocity, Reynolds stress and k contour maps, normalized by U_o
3. circumferential distribution of C_f

In the symmetry plane, along the lateral walls:

1. C_p distributions

Previous Numerical Studies

Sotiropoulos and Patel (ref 3.) have performed calculations of this case with the two-layer k- model using two different numerical methods: the "finite-analytic" and a finite-difference method. In both cases, the overall structure of the flow is well predicted but both the strength of the secondary motion, and consequently its effect on the streamwise flow development, and the effects of wall curvature on the turbulence within the lateral boundary layers are underestimated.

Main References

Description of experiments

1. KIM, W.J. & PATEL, V.C. (1994). Origin and decay of longitudinal vortices in developing flow in a curved rectangular duct. J. of Fluids Engineering, 116, 45.

Previous numerical studies

2. SOTIROPOULOS, F. & PATEL, V.C. (1992). Flow in curved ducts of varying cross-section. Institute of Hydraulic Research, University of Iowa, IIHR Report No. 358.