김경종
458 lines
31 KiB
Markdown
458 lines
31 KiB
Markdown
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apply to the more general functional of Eq. 16.4-3 and are summarized as follows.
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We write a temperature field T in terms of element nodal temperatures $\{T_{e}\}$ , and from it compute the required temperature gradients $\{T_{e}\}$ :
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$$
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T = \left\lfloor \mathrm{N} \right\rfloor \left\{\mathrm{T} _ {e} \right\} \quad \text { and } \quad \left\{\mathrm{T} _ {\partial} \right\} = [ \mathrm{B} ] \left\{\mathrm{T} _ {e} \right\} \quad \text { where } \quad [ \mathrm{B} ] = \{\partial \} [ \mathrm{N} ] \tag {16.5-1}
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$$
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but $(1 / r)$ is deleted from row 1 of $\{\partial\}$ if Eq. 16.4-6 is used. Since $T = T^T$ , $T^2 = T^T T$ , and $\dot{T} = [\mathbf{N}]\{\dot{\mathbf{T}}_e\}$ , Eq. 16.4-3 can be written, for one element,
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$$
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\Pi_ {e} = \frac {1}{2} \left\{\mathbf {T} _ {e} \right\} ^ {T} ([ \mathbf {k} ] + [ \mathbf {h} ]) \left\{\mathbf {T} _ {e} \right\} + \left\{\mathbf {T} _ {e} \right\} ^ {T} \left([ \mathbf {c} ] \left\{\dot {\mathbf {T}} _ {e} \right\} - \left\{\mathbf {r} _ {q} \right\} - \left\{\mathbf {r} _ {Q} \right\} - \left\{\mathbf {r} _ {h} \right\}\right) \tag {16.5-2}
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$$
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where
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$$
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[ \mathbf {k} ] = \int_ {V _ {e}} [ \mathbf {B} ] ^ {T} [ \boldsymbol {\kappa} ] [ \mathbf {B} ] d V \quad \{\mathbf {r} _ {q} \} = \int_ {S _ {e}} [ \mathbf {N} ] ^ {T} q _ {B} d S
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$$
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$$
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[ \mathbf {h} ] = \int_ {S _ {e}} [ \mathbf {N} ] ^ {T} h [ \mathbf {N} ] d S \quad \{\mathbf {r} _ {Q} \} = \int_ {V _ {e}} [ \mathbf {N} ] ^ {T} Q d V \tag {16.5-3}
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$$
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$$
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[ \mathbf {c} ] = \int_ {V _ {e}} [ \mathbf {N} ] ^ {T} \rho c [ \mathbf {N} ] d V \quad \{\mathbf {r} _ {h} \} = \int_ {S _ {e}} [ \mathbf {N} ] ^ {T} h T _ {f} d S
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$$
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By adding the $\Pi_{e}$ contributions of the elements, we obtain $\Pi$ of the assembled system. Assembly implies the usual expansion of element arrays to “structure size,” so that the global nodal temperature array $\{T\}$ replaces $\{T_{e}\}$ of each element, $[K] = \Sigma [k]$ , and so on. Equations that make $\Pi$ stationary are $\{\partial\Pi/\partial T\} = \{0\}$ , that is,
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$$
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([ \mathbf {K} ] + [ \mathbf {H} ]) \{\mathbf {T} \} + [ \mathbf {C} ] \{\dot {\mathbf {T}} \} = \{\mathbf {R} _ {q} \} + \{\mathbf {R} _ {Q} \} + \{\mathbf {R} _ {h} \} \tag {16.5-4}
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$$
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In a plane problem with convection heat transfer across a lateral surface, additional terms appear, as noted below Eq. 16.3-10. For convection across one lateral surface, we define
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$$
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[ \mathbf {h} _ {\mathrm{ls}} ] = \int \int \lfloor \mathbf {N} \rfloor^ {T} h \lfloor \mathbf {N} \rfloor d x d y \quad \text { and } \quad \{\mathbf {r} _ {\mathrm{ls}} \} = \int \int \lfloor \mathbf {N} \rfloor^ {T} h T _ {f} d x d y \tag {16.5-5}
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$$
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The terms $[H_{ls}]\{T\}$ and $\{R_{ls}\}$ must be added to the left- and right-hand sides, respectively, of Eq. 16.5-4.
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Remarks. If the medium is isotropic, $[\kappa]$ becomes the diagonal matrix $k[1\quad1\quad1]$ in general solids and solids of revolution, or $k[1\quad1]$ in plane problems and axisymmetric solids with axisymmetric temperature distribution.
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The simplest special form of Eq. 16.5-4 is $[K]\{T\} = \{0\}$ , which represents steady-state conditions without internal sources or sinks, some of the nodal temperatures $T_{i}$ in $\{T\}$ prescribed, and no heat flow across the boundary other than that implicitly associated with the prescribed $T_{i}$ .
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Prescribed nodal temperatures—and their spatial derivatives, if also used as nodal d.o.f. in $\{T\}$ —can be treated like prescribed nodal displacements in structural mechanics (Section 2.10). Thus, as in Fig. 2.10-7, the right-hand side of Eq. 16.5-4 is modified, and ones and zeros appear in the coefficient matrix. An alter-
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native procedure, which corresponds to Fig. 2.10-6, is to add a large conductivity $K_{D}$ to the appropriate diagonal coefficient in [K] and augment the corresponding coefficient on the right-hand side by $K_{D}\overline{T}$ , where $\overline{T}$ is the prescribed nodal temperature. This treatment greatly increases the maximum eigenvalues of the system and may therefore cause trouble in a direct integration analysis of how temperature varies with time [16.1].
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Matrices defined by Eqs. 16.5-3 and matrices used in structural mechanics have the following analogies of form:
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[k] is analogous to a conventional stiffness matrix.
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[h] is analogous to an elastic foundation stiffness matrix.
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[c] is analogous to a mass matrix.
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$\{r_{0}\}$ is analogous to nodal loads from body force.
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$\{\mathbf{r}_q\}, \{\mathbf{r}_h\}$ are analogous to nodal loads from surface traction.
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Matrix [c] is analogous to mass matrix [m] in that both multiply time derivatives of nodal d.o.f. and both provide resistance to time rates of change. But [c] multiplies first derivatives and [m] multiplies second derivatives.
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Like [m], [c] may be formulated as a consistent matrix or as a lumped matrix. The simplification provided by lumping may be accompanied by loss of accuracy. For an element with $n$ nodal temperatures in $\{\mathbf{T}_e\}$ , a lumped form of [c] results from multiplying $\rho c$ by the element volume $V_e$ and assigning $\rho cV_e / n$ to each element node. Thus $\rho cV_e / n$ appears $n$ times in a diagonal matrix [c]. Similar ad hoc lumping can be used to simplify calculation of [h] or $\{\mathbf{r}_Q\}$ . An $\{\mathbf{r}_Q\}$ or $\{\mathbf{R}_Q\}$ vector that contains a single nonzero term represents a point source or a point sink at a node. Proportional and optimal capacity-lumping schemes can be employed, in direct analogy to mass-lumping schemes described in Section 13.3, and having the same trade-offs of advantage and disadvantage.
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If the element is formulated in isoparametric fashion, then $dV = J \, d\xi \, d\eta \, d\zeta$ and the usual coordinate transformations are invoked. If the problem is essentially plane but the body is of varying thickness, then $dV = \tau \, dx \, dy$ , where thickness $\tau$ is a function of $x$ and $y$ . If the body is a solid of revolution, then $dV = r \, dr \, d\theta \, dz$ or $dV = rJ \, d\xi \, d\eta \, d\theta$ , in which radius $r$ should not be taken as constant over an element unless the element is far from the axis of revolution.
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Matrices [h], $\{\mathbf{r}_q\}$ , and $\{\mathbf{r}_h\}$ are zero unless the element has an edge or face on boundary $S$ and either convection or prescribed flux is associated with that part of $S$ . Even then, these matrices receive nonzero contributions only from the element edge or face that forms part of $S$ .
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It often happens that $[\kappa]$ must be regarded as a function of temperature. For steady-state conditions, a simple but perhaps inefficient way to solve this nonlinear problem is as follows. Estimate numerical values of conductivities, generate [K], and solve for {T}. Use these temperatures to obtain improved estimates of conductivities, generate a new [K], and solve for the new {T}. Repeat until convergence. Refinements of this iterative process are of course possible [16.2]. Techniques discussed in Chapter 17 can also be used.
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<details>
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<summary>text_image</summary>
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(boundary)
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Tf
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L34
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y
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η
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ξ
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4
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P
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1
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2
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x
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</details>
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Figure 16.5-1. Bilinear isoparametric element at the boundary of a plane structure where convection occurs.
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$T = [N]\{T_e\} = N_1T_1 + N_2T_2 + N_3T_3 + N_4T_4,$ where the $N_i$ are given by Eq. 6.3-2. Temperature gradients $\{T_d\}$ are
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$$
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\left\{ \begin{array}{l} T _ {, x} \\ T _ {, y} \end{array} \right\} = [ \mathrm{J} ] ^ {- 1} \left\{ \begin{array}{l} T _ {, \xi} \\ T _ {, \eta} \end{array} \right\} = \underbrace {[ \mathrm{J} ] ^ {- 1} \left[ \begin{array}{c c c c} N _ {1 , \xi} & N _ {2 , \xi} & N _ {3 , \xi} & N _ {4 , \xi} \\ N _ {1 , \eta} & N _ {2 , \eta} & N _ {3 , \eta} & N _ {4 , \eta} \end{array} \right]} _ {[ \mathrm{B} ]} \left\{ \begin{array}{l} T _ {1} \\ T _ {2} \\ T _ {3} \\ T _ {4} \end{array} \right\} \tag {16.5-6}
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$$
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where Jacobian matrix [J] is defined by Eq. 6.3-11. The conductivity matrix [k] for an element of thickness $\tau$ becomes
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$$
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[ \mathbf {k} ] = \int_ {- 1} ^ {1} \int_ {- 1} ^ {1} [ \mathbf {B} ] ^ {T} [ \boldsymbol {\kappa} ] [ \mathbf {B} ] \tau J d \xi d \eta \tag {16.5-7}
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$$
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where $\tau$ may be a function of $\xi$ and $\eta$ .
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Convection matrix [h] receives a contribution from side 3–4 only. Along side 3–4, $N_{1} = N_{2} = 0$ , $N_{3} = (1 + \xi)/2$ , $N_{4} = (1 - \xi)/2$ , and $J = L_{34}/2$ . Therefore, if $\tau$ and h are independent of $\xi$ ,
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$$
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[ \mathbf {h} ] = \int_ {- 1} ^ {1} [ \mathbf {N} ] ^ {T} h [ \mathbf {N} ] \tau J d \xi = \frac {h \tau L _ {3 4}}{6} \left[ \begin{array}{l l l l} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 2 & 1 \\ 0 & 0 & 1 & 2 \end{array} \right] \tag {16.5-8}
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$$
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In similar fashion, for $\{r_{h}\}$ we obtain
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$$
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\{\mathbf {r} _ {h} \} = \int_ {- 1} ^ {1} \left| \mathbf {N} \right| ^ {T} h T _ {f} \tau J d \xi = T _ {f} \frac {h \tau L _ {3 4}}{2} \left\{ \begin{array}{l} 0 \\ 0 \\ 1 \\ 1 \end{array} \right\} \tag {16.5-9}
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$$
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If a heat input $Q_{P}$ (units J/s) is prescribed at point P, we obtain the resulting $\{r_{Q}\}$ vector from Eq. 16.5-3 by saying that Q = 0 except at point P,
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$$
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\{\mathbf {r} _ {Q} \} = \int_ {V _ {\epsilon}} \left[ \mathbf {N} \right] ^ {T} Q d V = \left[ \mathbf {N} \right] _ {P} ^ {T} \int_ {V _ {\epsilon}} Q d V = \left[ \mathbf {N} \right] _ {P} ^ {T} Q _ {P}. \tag {16.5-10}
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$$
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where $\left[N\right]_{P}$ is the value of $\left[N\right]$ at point P.
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# 16.6 THERMAL TRANSIENTS
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In thermal problems, as in structural mechanics, a time-varying solution may be obtained by the modal method or by direct temporal integration. If material properties are not temperature-dependent, and the solution is dominated by a few of the lowest eigenmodes and is needed over a long time span, then the modal method is favored. If the problem is nonlinear, or the solution displays sharp transients (which require many eigenmodes for an accurate description) and is needed over a short time span, then direct integration is favored.
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By either method, the equations to be solved have the form
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$$
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[ \mathbf {K} _ {T} ] \{\mathbf {T} \} + [ \mathbf {C} ] \{\dot {\mathbf {T}} \} = \{\mathbf {R} \} \tag {16.6-1}
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$$
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This equation is the same as Eq. 16.5-4, in which $[K_{T}]$ contains all matrices that premultiply $\{T\}$ , and $\{R\}$ contains all vectors on the right-hand side. One seeks to determine $\{T\}$ as a function of time when $[K_{T}]$ and $[C]$ are known, $\{R\}$ is a known function of time, and the initial temperatures at time t = 0 are known. Unless stated otherwise, we presume that $[K_{T}]$ and $[C]$ are independent of $\{T\}$ .
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Modal Method. The procedure is very similar to that used for structural dynamics (Section 13.6). It is outlined as follows. One first considers the eigenproblem
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$$
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([ \mathbf {K} _ {T} ] - \lambda [ \mathbf {C} ]) \{\overline {{\mathbf {T}}} \} = \{\mathbf {0} \} \tag {16.6-2}
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$$
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If each eigenvector $\{\overline{\mathbf{T}}\}_i$ is normalized with respect to $[\mathbf{C}]$ , that is, if $\{\overline{\mathbf{T}}\}_i^T [\mathbf{C}]\{\overline{\mathbf{T}}\}_i = 1$ , then
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$$
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[ \phi ] ^ {T} [ \mathbf {C} ] [ \phi ] = [ \mathbf {I} ] \quad \text { and } \quad [ \phi ] ^ {T} [ \mathbf {K} _ {T} ] [ \phi ] = [ \lambda ] \tag {16.6-3}
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$$
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where $[\phi]$ is the modal matrix; that is, a matrix whose columns are the normalized eigenvectors $\{\overline{\mathbf{T}}\}_{1}, \{\overline{\mathbf{T}}\}_{2}$ , and so on, $[\mathbf{I}]$ is a unit matrix, and $[\lambda]$ is the (diagonal) spectral matrix $[\lambda] = [\lambda_1 - \lambda_2 \ldots \lambda_n]$ . Nodal temperatures are transformed to generalized temperatures $\{\mathbf{Z}\}$ by
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$$
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\{\mathbf {T} \} = [ \phi ] \{\mathbf {Z} \} \tag {16.6-4}
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$$
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where the $Z_{i}$ in $\{Z\}$ state the proportion of each eigenvector in the transformation. We substitute Eq. 16.6-4 into Eq. 16.6-1, premultiply by $[\phi]^{T}$ , and take note of Eqs. 16.6-3. Thus we obtain uncoupled equations, each having the form
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$$
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\dot {Z} _ {i} + \lambda_ {i} Z _ {i} = p _ {i} \quad \text { where } \quad p _ {i} = \{\phi \} _ {i} ^ {T} \{\mathbf {R} \} \tag {16.6-5}
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$$
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Here $\{\phi\}_{i}$ is the $i$ th column of $[\phi]$ , and $i$ runs from 1 to $m$ , where $m$ is typically much less than the total number of d.o.f.; that is, only the first few columns of $[\phi]$ are used. After Eqs. 16.6-5 are integrated, $\{\mathbf{Z}\} = \{\mathbf{Z}(t)\}$ is known, and Eq. 16.6-4 yields $\{\mathbf{T}\} = \{\mathbf{T}(t)\}$ .
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Variants of modal methods used in structural mechanics can also be applied to thermal transient analysis [16.3].
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Direct Integration. Consider two temperature states, separated by time increment
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$\Delta t$ , and denoted by $\{T\}_{n}$ and $\{T\}_{n+1}$ . A temporal integration scheme, known as the generalized trapezoidal rule, is based on the assumption that the two temperature states have the relation
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$$
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\{\mathbf {T} \} _ {n + 1} = \{\mathbf {T} \} _ {n} + \{(1 - \beta) \dot {\mathbf {T}} _ {n} + \beta \dot {\mathbf {T}} _ {n + 1} \} (\Delta t) \tag {16.6-6}
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$$
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Like Newmark's method for the second-order equations of structural dynamics, Eq. 16.6-6 contains a factor $\beta$ that the analyst may select. Next we write Eq. 16.6-1 for time $t$ and again for time $t + \Delta t$ , then multiply the first equation by $1 - \beta$ and the second by $\beta$ . Thus
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$$
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(1 - \beta) \left([ \mathbf {K} _ {T} ] \{\mathbf {T} \} _ {n} + [ \mathbf {C} ] \{\dot {\mathbf {T}} \} _ {n}\right) = (1 - \beta) \{\mathbf {R} \} _ {n} \tag {16.6-7a}
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$$
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$$
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\beta \left(\left[ \mathbf {K} _ {T} \right] \{\mathbf {T} \} _ {n + 1} + [ \mathbf {C} ] \{\dot {\mathbf {T}} \} _ {n + 1}\right) = \beta \{\mathbf {R} \} _ {n + 1} \tag {16.6-7b}
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$$
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Equations 16.6-7 are added, then Eq. 16.6-6 is used to eliminate time derivatives of temperature. This step requires that [C] not change with time. The result is
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$$
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\begin{array}{l} \left(\frac {1}{\Delta t} [ \mathbf {C} ] + \beta [ \mathbf {K} _ {T} ]\right) \{\mathbf {T} \} _ {n + 1} = \left(\frac {1}{\Delta t} [ \mathbf {C} ] - (1 - \beta) [ \mathbf {K} _ {T} ]\right) \{\mathbf {T} \} _ {n} \\ + (1 - \beta) \{\mathbf {R} \} _ {n} + \beta \{\mathbf {R} \} _ {n + 1} \tag {16.6-8} \\ \end{array}
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$$
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From a known $\{T\}_{0}$ at t = 0, Eq. 16.6-8 yields $\{T\}_{1}$ at $t = \Delta t$ . Then, using $\{T\}_{1}$ , we determine $\{T\}_{2}$ at $t = 2(\Delta t)$ , and so on. If $\Delta t$ is not changed, the coefficient of $\{T\}_{n+1}$ need be generated and forward-reduced only once; the equations are then repeatedly solved for a sequence of right-hand sides.
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Depending on $\beta$ , time step $\Delta t$ in Eq. 16.6-8 may have an upper limit if the algorithm is to be numerically stable. If $\beta < \frac{1}{2}$ the largest $\Delta t$ for stability is [16.4]
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$$
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\Delta t _ {\mathrm{cr}} = \frac {2}{(1 - 2 \beta) \lambda_ {\max}} \tag {16.6-9}
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$$
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where $\lambda_{max}$ is the largest eigenvalue of Eq. 16.6-2. If $\beta \geq \frac{1}{2}$ the algorithm is unconditionally stable; that is, stability (but not accuracy) is guaranteed as $\Delta t$ becomes indefinitely large. Names associated with various schemes are as follows:
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$$
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\begin{array}{l} \beta = 0 \quad \text { forward difference or Euler (conditionally stable) } \\ \beta = \frac {1}{2} \quad \text { Crank - - Nicolson or trapezoidal rule (unconditionally stable) } \\ \beta = \frac {2}{3} \quad \text { Galerkin (unconditionally stable) } \\ \beta = 1 \quad \text { backward difference (unconditionally stable) } \\ \end{array}
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$$
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If $\beta = 0$ , the algorithm is termed explicit. If $\beta > 0$ , it is termed implicit. If [C] is a diagonal matrix and $\beta = 0$ , the computational effort per time step is small but so is $\Delta t_{\mathrm{cr}}$ . Among implicit methods, the choice $\beta = \frac{1}{2}$ is popular, but sharp transients may excite annoying oscillations in the solution. Oscillations can be reduced by using a smaller value of $\Delta t$ or numerically damped by using a value of $\beta$ somewhat greater than $\frac{1}{2}$ . If the problem is nonlinear the only unconditionally stable form of Eq. 16.6-8 is $\beta = 1$ ; however, it is not particularly accurate [16.4].
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Various other direct integration algorithms are available [16.5]. Detailed dis-
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cussion of selected time-stepping algorithms may be found in the latter sections of Chapter 13.
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# 16.7 RELATED PROBLEMS. FLUID FLOW
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Several physical phenomena are described by the same form of differential equation that describes heat conduction, Eq. 16.3-5 or 16.4-1. For steady-state conditions and a homogeneous and isotropic material, this equation has the form
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$$
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k \nabla^ {2} \phi + Q = 0 \tag {16.7-1}
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$$
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where $\nabla^{2}$ is the Laplacian operator (defined in Eq. 16.7-4). Phenomena described by Eq. 16.7-1, or by its less specialized form, include the following:
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Heat conduction ( $\phi = \text{temperature}$ )
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Viscous flow in a pipe ( $\phi = \text{axial velocity}$ )
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Groundwater flow ( $\phi = \text{hydraulic head}$ )
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Pressurized membrane ( $\phi = \text{membrane deflection}$ )
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Elastic torsion ( $\phi = \text{stress function or warping function}$ )
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Electric conduction ( $\phi = \text{voltage}$ )
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Electrostatics ( $\phi = \text{field potential}$ )
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Magnetostatics ( $\phi = \text{magnetic potential}$ )
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Potential flow ( $\phi = \text{velocity potential or stream function}$ )
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For the first four phenomena cited, k represents conductivity, viscosity, permeability, and surface tension, and Q represents internal heat generated, pressure gradient, flow associated with a source or a sink, and pressure, respectively.
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With appropriate definition of variables, material properties, and boundary conditions, problems in any of these areas can be addressed by use of the computational procedures already established for heat conduction analysis. Indeed, several of these problems can be solved by appropriate use of a computer program for structural analysis [16.6]. For example, if a viscous incompressible fluid flows so slowly that inertia effects are negligible, the two-dimensional problem is described by the equation $\nabla^4\psi = 0$ , where $\psi$ is the stream function defined in Eq. 16.7-3b [16.7]. This equation has the same form as the equation $\nabla^4 w = q / D$ that describes bending of a flat plate. It is highly recommended that use of any of these analogies be preceded by study of the specific problem area in question.
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Potential Flow. A particularly simple example of Eq. 16.7-1 is provided by plane potential flow—that is, irrotational flow of an incompressible and inviscid fluid. Let u and v represent flow velocities in the x and y directions, respectively. The conditions of irrotationality and continuity are, respectively,
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$$
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u _ {, y} - v _ {, x} = 0 \qquad \text { and } \qquad u _ {, x} + v _ {, y} = 0 \tag {16.7-2}
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$$
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For analysis, we can use either a potential function $\phi = \phi(x, y)$ or a stream function $\psi = \psi(x, y)$ , which are defined such that
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$$
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\text { Potential function } \phi \text {:} u = \phi_ {, x} \quad \text { and } \quad v = \phi_ {, y} \tag {16.7-3a}
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$$
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$$
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\text { Stream function } \psi : \quad u = \psi_ {, y} \quad \text { and } \quad v = - \psi_ {, x} \tag {16.7-3b}
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$$
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If n and s are arbitrarily oriented, right-handed, orthogonal coordinates in the xy plane, then $\phi_{,n}$ represents flow velocity in the positive n direction and $\psi_{,n}$ represents flow velocity in the negative s direction (see Fig. 16.7-1b).
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Substitution of Eqs. 16.7-3 into Eqs. 16.7-2 shows that one of the two equations becomes $0 \equiv 0$ while the other becomes Laplace's equation. Thus the problem is described by
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$$
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\nabla^ {2} \phi = 0 \quad \text { or by } \quad \nabla^ {2} \psi = 0 \quad \text { where } \quad \nabla^ {2} = \frac {\partial^ {2}}{\partial x ^ {2}} + \frac {\partial^ {2}}{\partial y ^ {2}} \tag {16.7-4}
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$$
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A finite element formulation produces element matrices of the form
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$$
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[ \mathbf {k} ] \{\phi_ {e} \} = \{\mathbf {r} \} \quad \text { or } \quad [ \mathbf {k} ] \{\psi_ {e} \} = \{\mathbf {r} \} \tag {16.7-5}
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$$
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where $\{\phi_{e}\}$ and $\{\psi_{e}\}$ contain nodal values of $\phi$ and $\psi$ , respectively, [k] is obtained from Eqs. 16.5-3 with $[\kappa]$ a unit matrix and dV = (1) dx dy, and $\{r\}$ comes from $\{r_{q}\}$ of Eqs. 16.5-3 with $q_{B}$ representing the prescribed flow rate $\phi_{,n}$ or $\psi_{,s}$ in a direction n normal to boundary S.
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|
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An example application is depicted in Fig. 16.7-1. Uniform flow at velocity $u_{0}$ enters at the left. Flow velocities in the neighborhood of the cylindrical obstacle are desired. Because of symmetry about horizontal and vertical centerlines, only one quadrant need be modeled. Other known symmetries of the flow pattern, when associated with Eqs. 16.7-3, dictate boundary conditions shown in Fig. 16.7-1. For example, v = 0 along AB, CD, and DE; therefore, $\phi_{,y} = 0$ along these
|
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|
||
|
||
|
||
<details>
|
||
<summary>text_image</summary>
|
||
|
||
u₀
|
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Quadrant modeled
|
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H
|
||
H
|
||
</details>
|
||
|
||
(a)
|
||
|
||
|
||
|
||
<details>
|
||
<summary>text_image</summary>
|
||
|
||
y
|
||
φ,y = 0
|
||
φ,x = u₀
|
||
E
|
||
D
|
||
φ = 0
|
||
s
|
||
C
|
||
A
|
||
B
|
||
n
|
||
φ,n = 0
|
||
φ,y = 0
|
||
x
|
||
</details>
|
||
|
||
(b)
|
||
|
||
|
||
|
||
<details>
|
||
<summary>text_image</summary>
|
||
|
||
ψ = H
|
||
ψ = u₀y
|
||
ψ = 0
|
||
ψ = 0
|
||
ψ = x = 0
|
||
ψ = 0
|
||
x
|
||
y
|
||
A
|
||
B
|
||
C
|
||
D
|
||
</details>
|
||
|
||
(c)
|
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Figure 16.7-1. (a) Flow around a cylindrical obstacle in a rectangular channel. (b,c) Boundary conditions associated with potential function and stream function solutions, respectively, in the quadrant modeled.
|
||
|
||
<!-- source-page: 508 -->
|
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|
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lines. The fluid has no velocity normal to the cylinder; therefore $\phi_{,n} = 0$ along $BC$ . A nodal value of $\phi$ must be assigned in order to make the coefficient matrix nonsingular, but the numerical value is arbitrary because only derivatives of $\phi$ are of interest. If $\phi = 0$ is prescribed at $C$ , the value $\phi = 0$ is dictated all along $CD$ because of the condition $\phi_{,y} = 0$ along $CD$ . Analogous remarks apply to use of $\psi$ rather than $\phi$ . At points such as $D$ , both essential and nonessential boundary conditions appear. Here one may prefer to assign $\phi_D = 0$ or $\psi_D = H$ rather than assigning the load terms $r_D = 0$ and letting $\phi_D$ or $\psi_D$ remain an unknown to be calculated by solving the global equivalent of Eqs. 16.7-5. If the formulation uses $\phi$ (or $\psi$ ) and its $x$ and $y$ derivatives as nodal d.o.f., values of $\phi$ and a derivative of $\phi$ may be applied at a single node, although not to a single d.o.f. at that node. When all nodal d.o.f. are known, gradients are calculated (e.g., by Eq. 16.5-6 with $T$ replaced by $\phi$ or $\psi$ ), and Eq. 16.7-3 yields the required flow velocities.
|
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|
||
# 16.8 FLUID VIBRATION AND WAVES, PRESSURE FORMULATION
|
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|
||
The governing differential equation, which is Eq. 16.8-4 in the discussion that follows, is called the wave equation. It describes phenomena in which energy is propagated by waves and has applications in problems of sound propagation, the sloshing of liquid in a container, and fluid-structure interaction. Its finite element expression has nodal pressures as d.o.f.
|
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|
||
Formulation. We consider a fluid without viscosity. Part of the total pressure at any point may be a hydrostatic pressure. The remaining pressure, denoted by p, is associated with motion of the fluid. Pressure gradients $p_{,x}$ , $p_{,y}$ , and $p_{,z}$ may exist in the respective coordinate directions. Hence, if Newton's law F = ma is applied to a differential element of volume dx dy dz, we obtain
|
||
|
||
$$
|
||
p _ {, x} = - \rho \ddot {u} \quad p _ {, y} = - \rho \ddot {v} \quad p _ {, z} = - \rho \ddot {w} \tag {16.8-1}
|
||
$$
|
||
|
||
where u, v, and w are displacements in the x, y, and z directions, and $\rho$ is the mass density, which is assumed to be essentially constant despite compressibility of the fluid. We differentiate each of Eqs. 16.8-1 with respect to its own spatial coordinate, then add. Thus
|
||
|
||
$$
|
||
p _ {, x x} + p _ {, y y} + p _ {, z z} = - \rho (i i _ {, x} + \ddot {v} _ {, y} + \ddot {w} _ {, z}) \tag {16.8-2a}
|
||
$$
|
||
|
||
or
|
||
|
||
$$
|
||
\nabla^ {2} p = - \rho \frac {d ^ {2}}{d t ^ {2}} \left(\epsilon_ {x} + \epsilon_ {y} + \epsilon_ {z}\right) \tag {16.8-2b}
|
||
$$
|
||
|
||
where $\epsilon_{x} = u_{,x}$ , and so on. Bulk modulus $B$ is defined as
|
||
|
||
$$
|
||
B = - \frac {p}{d V / V} = - \frac {p}{\epsilon_ {x} + \epsilon_ {y} + \epsilon_ {z}} \tag {16.8-3}
|
||
$$
|
||
|
||
The negative sign appears because volume decreases under increasing pressure. Equations 16.8-2b and 16.8-3 yield
|
||
|
||
<!-- source-page: 509 -->
|
||
|
||
$$
|
||
\nabla^ {2} p = \frac {\rho}{B} \ddot {p} \quad \text { or } \quad \nabla^ {2} p = \frac {1}{c ^ {2}} \ddot {p} \tag {16.8-4}
|
||
$$
|
||
|
||
where c is the speed of sound in the medium, $c = \sqrt{B/\rho}$ . Equation 16.8-4 is the wave equation.
|
||
|
||
Equation 16.8-4 must be solved in a volume V, subject to boundary conditions on its surface S. The essential boundary condition is p = 0, which prevails on a free fluid surface if waves are negligible. The nonessential boundary condition, which prevails on a solid boundary, is
|
||
|
||
$$
|
||
\frac {\partial p}{\partial n} = - \rho \ddot {u} _ {n} \tag {16.8-5}
|
||
$$
|
||
|
||
where n = outward normal direction and $\ddot{u}_{n} = \text{acceleration of the boundary in direction } n$ . For a rigid boundary, $\ddot{u}_{n} = 0$ . If the effect of small-amplitude waves on a fluid surface is to be included, we write $p = \rho g w_{s}$ , where g = acceleration of gravity and $w_{s} = \text{surface elevation relative to the mean surface level}$ . Combining $p = \rho g w_{s}$ with $p_{,z} = -\rho \ddot{w}_{s}$ from Eqs. 16.8-1, we obtain
|
||
|
||
$$
|
||
\frac {\partial p}{\partial z} = - \frac {1}{g} \ddot {p} \tag {16.8-6}
|
||
$$
|
||
|
||
as the boundary condition on a fluid surface with small-amplitude waves.
|
||
|
||
A functional for this problem is
|
||
|
||
$$
|
||
\Pi = \int_ {V} \left(\frac {p _ {, x} ^ {2} + p _ {, y} ^ {2} + p _ {, z} ^ {2}}{2} + \frac {\rho}{B} p \ddot {p}\right) d V + \int_ {S _ {s}} \rho \ddot {u} _ {n} p d S + \int_ {S _ {f}} \frac {1}{g} p \ddot {p} d S \tag {16.8-7}
|
||
$$
|
||
|
||
where $S_{s} =$ solid boundary and $S_{f} =$ fluid boundary with wave action. As in Eqs. 16.8-4, $\rho / B$ may be replaced by $1 / c^2$ . The stationary condition $\delta \Pi = 0$ yields Eqs. 16.8-4, 16.8-5, and 16.8-6.
|
||
|
||
A finite element formulation follows the familiar pattern. Pressure p within an element is interpolated from nodal d.o.f. $\{P_{e}\}$ , where $\{P_{e}\}$ may contain nodal pressures only or may also contain spatial derivatives of nodal pressures. Thus
|
||
|
||
$$
|
||
p = \lfloor \mathrm{N} \rfloor \{\mathrm{P} _ {e} \} \quad \text { and } \quad \ddot {p} = \lfloor \mathrm{N} \rfloor \{\ddot {\mathrm{P}} _ {e} \} \tag {16.8-8}
|
||
$$
|
||
|
||
We define the following global matrices, where summation signs indicate assembly of element matrices. The notation is analogous to that in Eqs. 16.5-3.
|
||
|
||
$$
|
||
[ \mathbf {K} ] = \sum \int_ {V _ {e}} \left(\left[ \mathbf {N}, _ {x} \right] ^ {T} \left[ \mathbf {N}, _ {x} \right] + \left[ \mathbf {N}, _ {y} \right] ^ {T} \left[ \mathbf {N}, _ {y} \right] + \left[ \mathbf {N}, _ {z} \right] ^ {T} \left[ \mathbf {N}, _ {z} \right]\right) d V \tag {16.8-9a}
|
||
$$
|
||
|
||
$$
|
||
[ \mathbf {C} ] = \sum \int_ {V _ {e}} \frac {\rho}{B} [ \mathbf {N} ] ^ {T} [ \mathbf {N} ] d V \tag {16.8-9b}
|
||
$$
|
||
|
||
$$
|
||
[ \mathbf {H} ] = \sum \int_ {S _ {f}} [ \mathbf {N} ] ^ {T} [ \mathbf {N} ] \frac {1}{g} d S \quad \left\{\mathbf {R} _ {s} \right\} = \sum \int_ {S _ {s}} [ \mathbf {N} ] ^ {T} \rho \ddot {u} _ {n} d S \tag {16.8-9c}
|
||
$$
|
||
|
||
<!-- source-page: 510 -->
|
||
|
||
Equation 16.8-7 becomes
|
||
|
||
$$
|
||
\Pi = \frac {1}{2} \{\mathbf {P} \} ^ {T} [ \mathbf {K} ] \{\mathbf {P} \} + \{\mathbf {P} \} ^ {T} ([ \mathbf {C} ] + [ \mathbf {H} ]) \{\ddot {\mathbf {P}} \} + \{\mathbf {P} \} ^ {T} \{\mathbf {R} _ {s} \} \tag {16.8-10}
|
||
$$
|
||
|
||
where $\{\mathbf{P}\}$ is the global array of nodal d.o.f. The stationary condition $\{\partial \Pi / \partial \mathbf{P}\} = \{\mathbf{0}\}$ yields the finite element formulation
|
||
|
||
$$
|
||
[ \mathbf {K} ] \{\mathbf {P} \} + ([ \mathbf {C} ] + [ \mathbf {H} ]) \{\ddot {\mathbf {P}} \} = - \{\mathbf {R} _ {s} \} \tag {16.8-11}
|
||
$$
|
||
|
||
One must often deal with a boundary at infinity, that is, a boundary so distant that reflected waves do not appear for the duration of the event of interest. Such a condition can be modeled by “infinite elements” or by other analytical procedures [16.9–16.11].
|
||
|
||
Sloshing and Acoustic Modes. Let walls that support or contain a fluid be rigid, so that $\ddot{u}_{n}=0$ . Thus the forcing function becomes zero, and the fluid vibrates in one of its natural modes. The pressure becomes $p=\overline{p}\sin\omega t$ , where $\omega$ is a natural frequency and amplitude $\overline{p}$ is a function of the spatial coordinates but is independent of time t. Equation 16.8-4 becomes
|
||
|
||
$$
|
||
\nabla^ {2} \overline {{{p}}} + \omega^ {2} \frac {\rho}{B} \overline {{{p}}} = 0 \quad \text { or } \quad \nabla^ {2} \overline {{{p}}} + \omega^ {2} \frac {1}{c ^ {2}} \overline {{{p}}} = 0 \tag {16.8-12}
|
||
$$
|
||
|
||
Similarly, a finite element formulation appears from Eq. 16.8-11 if we set $\{\mathbf{R}_s\} = \{\mathbf{0}\}$ and $\{\mathbf{P}\} = \{\overline{\mathbf{P}}\}\sin \omega t$ :
|
||
|
||
$$
|
||
[ [ \mathbf {K} ] - \omega^ {2} ([ \mathbf {C} ] + [ \mathbf {H} ]) ] \{\overline {{{\mathbf {P}}}} \} = \{\mathbf {0} \} \tag {16.8-13}
|
||
$$
|
||
|
||
This eigenvalue problem can be solved to obtain natural frequencies $\omega_{i}$ and the corresponding pressure modes $\{\overline{\mathbf{P}}\}_{i}$ . If $[\mathbf{C}] = [\mathbf{0}]$ , Eq. 16.8-13 yields the slosh frequencies of an incompressible liquid in a rigid container. If $[\mathbf{H}] = [\mathbf{0}]$ , Eq. 16.8-13 yields the acoustic modes of a fluid in a cavity with rigid walls.
|
||
|
||
As a special case, consider the one-dimensional problem of acoustic modes in a pipe with rigid walls that lies along an $x$ axis (Fig. 16.8-1). Thus, in Eq. 16.8-7, $p_{,y} = p_{,z} = 0$ and $\ddot{u}_n = 0$ . Volume $dV$ becomes $A dx$ , where $A = A(x)$ for a pipe of variable cross section. The integral over $S_f$ is discarded because there are no surface waves, and the integral over $S_s$ is zero because $\ddot{u}_n = 0$ . Thus, only special forms of Eqs. 16.8-9a and 16.8-9b are used in Eq. 16.8-13. As an alternative to specialization, one may begin with a functional for this particular problem. It is
|
||
|
||
$$
|
||
\Pi = \int \left(\overline {{p}} _ {, x} ^ {2} - \frac {\omega^ {2}}{c ^ {2}} \overline {{p}} ^ {2}\right) A d x \tag {16.8-14}
|
||
$$
|
||
|
||
where $\overline{p}$ is the pressure amplitude. By making the usual definitions and substitutions (analogous to Eqs. 16.8-8 and 16.8-9), we may develop a finite element formulation directly from Eq. 16.8-14 [16.12]. As for boundary conditions: at a closed end, $\overline{p}_{,x}=0$ and $\overline{p}$ is unknown; at an open end, $\overline{p}=0$ and $\overline{p}_{,x}$ is unknown.
|