In
fluid dynamics, stagnation point flow represents the flow of a fluid in the immediate neighborhood of a stagnation point (or a stagnation line) with which the stagnation point (or the line) is identified for a potential flow or inviscid flow. The flow specifically considers a class of stagnation points known as saddle points where the incoming streamlines gets deflected and directed outwards in a different direction; the streamline deflections are guided by separatrices. The flow in the neighborhood of the stagnation point or line can generally be described using
potential flow
In fluid dynamics, potential flow (or ideal flow) describes the velocity field as the gradient of a scalar function: the velocity potential. As a result, a potential flow is characterized by an irrotational velocity field, which is a valid app ...
theory, although viscous effects cannot be neglected if the stagnation point lies on a solid surface.
Stagnation point flow without solid surfaces
When two streams either of two-dimensional or axisymmetric nature impinge on each other orthogonally, a stagnation plane is created, where the incoming streams are diverted tangentially outwards; thus on the stagnation plane, the velocity component normal to that plane is zero, whereas the tangential component is non-zero. In the neighborhood of the stagnation point, a local description for the velocity field can be described.
General three-dimensional velocity field
The stagnation point flow corresponds to a linear dependence on the coordinates, that can be described in the Cartesian coordinates
with velocity components
as follows
:
where
are constants referred as the strain rates; these constants are not completely arbitrary since the continuity equation requires
, that is to say, only two of the three constants are independent. We shall assume
so that flow is towards the stagnation point in the
direction and away from the stagnation point in the
direction. Without loss of generality, one can assume that
. The flow field can be categorized into different types based on a single parameter
:
Planar stagnation-point flow
The two-dimensional stagnation-point flow belongs to the case
. The flow field is described as follows
:
where we let
. This flow field is investigated as early as 1934 by
G. I. Taylor
Sir Geoffrey Ingram Taylor OM FRS FRSE (7 March 1886 – 27 June 1975) was a British physicist and mathematician, and a major figure in fluid dynamics and wave theory. His biographer and one-time student, George Batchelor, described him as ...
. In the laboratory, this flow field is created using a four-mill apparatus, although these flow fields are ubiquitous in turbulent flows.
Axisymmetric stagnation-point flow
The axisymmetric stagnation point flow corresponds to
. The flow field can be simply described in cylindrical coordinate system
with velocity components
as follows
:
where we let
.
Radial stagnation flows
In radial stagnation flows, instead of a stagnation point, we have a stagnation circle and the stagnation plane is replaced by a stagnation cylinder. The radial stagnation flow is described using the cylindrical coordinate system
with velocity components
as follows
:
where
is the location of the stagnation cylinder.
Hiemenz flow
The flow due to the presence of a solid surface at
in planar stagnation-point flow was described first by Karl Hiemenz in 1911, whose numerical computations for the solutions were improved later by
Leslie Howarth
Leslie Howarth FRS (23 May 1911 – 22 September 2001) was a British mathematician who dealt with hydrodynamics and aerodynamics.
Biography
Howarth was educated at Accrington Academy#Accrington Grammar School, Accrington Grammar School, from wh ...
. A familiar example where Hiemenz flow is applicable is the forward stagnation line that occurs in the flow over a circular cylinder.
The solid surface lies on the
. According to potential flow theory, the fluid motion described in terms of the
stream function
The stream function is defined for incompressible ( divergence-free) flows in two dimensions – as well as in three dimensions with axisymmetry. The flow velocity components can be expressed as the derivatives of the scalar stream function. T ...
and the velocity components
are given by
:
The stagnation line for this flow is
. The velocity component
is non-zero on the solid surface indicating that the above velocity field do not satisfy no-slip boundary condition on the wall. To find the velocity components that satisfy the no-slip boundary condition, one assumes the following form
:
where
is the
Kinematic viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the int ...
and
is the characteristic thickness where viscous effects are significant. The existence of constant value for the viscous effects thickness is due to the competing balance between the fluid convection that is directed towards the solid surface and viscous diffusion that is directed away from the surface. Thus the vorticity produced at the solid surface is able to diffuse only to distances of order
; analogous situations that resembles this behavior occurs in
asymptotic suction profile and
von Kármán swirling flow Von Kármán swirling flow is a flow created by a uniformly rotating infinitely long plane disk, named after Theodore von Kármán who solved the problem in 1921. The rotating disk acts as a fluid pump and is used as a model for centrifugal fans or ...
. The velocity components, pressure and
Navier–Stokes equations
In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...
then become
:
:
The requirements that
at
and that
as
translate to
:
The condition for
as
cannot be prescribed and is obtained as a part of the solution. The problem formulated here is a special case of
Falkner-Skan boundary layer. The solution can be obtained from numerical integrations and is shown in the figure. The asymptotic behaviors for large
are
:
where
is the
displacement thickness.
Stagnation point flow with a translating wall
Hiemenz flow when the solid wall translates with a constant velocity
along the
was solved by Rott (1956). This problem describes the flow in the neighbourhood of the forward stagnation line occurring in a flow over a rotating cylinder. The required stream function is
:
where the function
satisfies
:
The solution to the above equation is given by
Oblique stagnation point flow
If the incoming stream is perpendicular to the stagnation line, but approaches obliquely, the outer flow is not potential, but has a constant
vorticity
In continuum mechanics, vorticity is a pseudovector field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point and traveling along wi ...
. The appropriate stream function for oblique stagnation point flow is given by
:
Viscous effects due to the presence of a solid wall was studied by Stuart (1959), Tamada (1979) and Dorrepaal (1986). In their approach, the streamfunction takes the form
:
where the function
:
.
Homann flow
The solution for axisymmetric stagnation point flow in the presence of a solid wall was first obtained by Homann (1936). A typical example of this flow is the forward stagnation point appearing in a flow past a sphere.
Paul A. Libby (1974)(1976) extended Homann's work by allowing the solid wall to translate along its own plane with a constant speed and allowing constant suction or injection at the solid surface.
The solution for this problem is obtained in the cylindrical coorindate system
by introducing
:
where
is the translational speed of the wall and
is the injection (or, suction) velocity at the wall. The problem is axisymmetric only when
. The pressure is given by
:
The
Navier–Stokes equations
In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...
then reduce to
:
along with boundary conditions,
:
When
, the classical Homann problem is recovered.
Plane counterflows
Jets emerging from a slot-jets creates stagnation point in between according to potential theory. The flow near the stagnation point can by studied using self-similar solution. This setup is widely used in
combustion
Combustion, or burning, is a high-temperature exothermic redox chemical reaction between a fuel (the reductant) and an oxidant, usually atmospheric oxygen, that produces oxidized, often gaseous products, in a mixture termed as smoke. Combus ...
experiments. The initial study of impinging stagnation flows are due to C.Y. Wang.
[Wang, C. Y. "Impinging stagnation flows." The Physics of fluids 30.3 (1987): 915–917.]
Let two fluids with constant properties denoted with suffix
flowing from opposite direction impinge, and assume the two fluids are immiscible and the interface (located at
) is planar. The velocity is given by
:
where
are strain rates of the fluids. At the interface, velocities, tangential stress and pressure must be continuous.
Introducing the self-similar transformation,
:
:
results equations,
:
:
The no-penetration condition at the interface and free stream condition far away from the stagnation plane become
:
But the equations require two more boundary conditions. At
, the tangential velocities
, the tangential stress
and the pressure
are continuous. Therefore,
:
where
(from outer inviscid problem) is used. Both
are not known ''apriori'', but derived from matching conditions. The third equation is determine variation of outer pressure
due to the effect of viscosity. So there are only two parameters, which governs the flow, which are
:
then the boundary conditions become
:
.
References
{{Reflist
Fluid mechanics
Fluid dynamics