{ "cells": [ { "cell_type": "code", "execution_count": 2, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "remove-input" ] }, "outputs": [], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "%config InlineBackend.rc = {'figure.dpi': 300}\n", "%config InlineBackend.figure_format = 'svg'\n", "import sys\n", "sys.path.append(\"../py_figures\")\n", "import diagrams\n", "from myst_nb import glue\n", "import inter_fig_chap3" ] }, { "cell_type": "markdown", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "(content:chap3:combined_confined_unconfined)=\n", "# Combined confined/unconfined flow with areal recharge\n", "So far in this chapter, separate solutions have been derived for confined flow (or flow with constant transmissivity) and unconfined flow. In this section, a case is considered where the flow changes from confined flow to unconfined flow somewhere in the aquifer. Consider flow between two parallel rivers in a confined aquifer with thickness $H$ ({numref}`Figure %s `); the rivers are in direct contact with the aquifer. The rivers are a distance $L$ apart, and the areal recharge is uniform and equal to $N$. The stage $h_l$ in the left river is above the top of the aquifer, and the stage $h_r$ in the right river is below the top of the aquifer. Hence, groundwater flow is confined near the left river, is unconfined near the right river, and changes somewhere in the aquifer from confined to unconfined." ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "remove-input", "remove-output" ] }, "outputs": [ { "data": { "application/papermill.record/image/svg+xml": "\n\n\n \n \n \n \n 2025-11-25T13:50:51.602133\n image/svg+xml\n \n \n Matplotlib v3.10.1, https://matplotlib.org/\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n", "application/papermill.record/text/plain": "
" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "combflow3_3" } }, "output_type": "display_data" }, { "data": { "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", " \n", " 2025-11-25T13:50:51.651773\n", " image/svg+xml\n", " \n", " \n", " Matplotlib v3.10.1, https://matplotlib.org/\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "\n" ], "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "fig = diagrams.solution3_3()\n", "glue(\"combflow3_3\", fig, display=False)" ] }, { "cell_type": "markdown", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ " ```{glue:figure} combflow3_3\n", "---\n", "name: fig_combflow3.3\n", "---\n", "Combined confined/unconfined flow between two rivers.\n", "```" ] }, { "cell_type": "markdown", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "     One approach to solving this problem is to write one solution for the head in the confined part of the aquifer and a separate solution for the head in the unconfined part and then try to find the point where flow changes from confined to unconfined. This is possible, but the use of the discharge potential provides an easier way. The discharge potential was already defined for unconfined flow, but not yet for confined flow. For confined flow, the discharge vector may be written as\n", "\n", "$$\n", "Q_x=-kH\\frac{\\text{d}h}{\\text{d}x}=-\\frac{\\text{d}}{\\text{d}x}(kHh)=-\\frac{\\text{d}\\Phi}{\\text{d}x}\n", "$$(eq_disvec_3.3) \n", "\n", "where both $k$ and $H$ are constant. The discharge potential for confined flow is defined as\n", "\n", "$$\n", "\\Phi = kHh + C\n", "$$(eq_dispot_3.3)\n", "\n", "where $C$ is an arbitrary constant. When the flow is purely confined, the constant $C$ can be set to zero, but when the flow is confined in parts of the aquifer and unconfined in other parts of the aquifer, the constant $C$ must be picked differently, as explained in the following.\n", "
     Now that both confined flow and unconfined flow are formulated as potential flow, the discharge vector may be written as Eq. ({eq}`eq_disvs3.0`) , the governing differential equation is the Poisson equation ({eq}`eq_continuity3.0`), and the general solution for one-dimensional steady potential flow is Eq. ({eq}`eq_secdif3.0`). This means that the discharge potential is a continuous function of $x$, irrespective of whether the flow is confined or unconfined. The discharge potential must be converted to a head that is also a continuous function of $x$ (as the head cannot suddenly jump up or down without a physical barrier to flow). This is achieved by proper selection of the constant $C$ in Eq. ({eq}`eq_dispot_3.3`). At the point where the flow transitions from confined to unconfined, the head in the aquifer is equal to $z_b+H$. The potential at this point is referred to as the transition potential $\\Phi_t$ and may be computed from the potential for unconfined flow (Eq. {eq}`eq_disp3.0`) as\n", "\n", "\n", "$$\n", "\\Phi_t = \\tfrac{1}{2}kH^2\n", "$$(eq_transp1_3.3)\n", "\n", "The constant $C$ is chosen such that the discharge potential for confined flow yields the same transition potential\n", "\n", "$$\n", "\\Phi_t = kH(z_b + H) + C = \\tfrac{1}{2}kH^2\n", "$$(eq_transp2_3.3)\n", "\n", "so that\n", "\n", "$$\n", "C=-\\tfrac{1}{2}kH^2 - kHz_b\n", "$$(eq_C_3.3)\n", "\n", "     Once the discharge potential is computed as a function of $x$, it is converted to a head using\n", "the appropriate equation. If the flow is confined,\n", "\n", "$$\n", "h=\\frac{\\Phi-C}{kH} \\hskip 2em \\Phi\\ge\\Phi_t\n", "$$(eq_headc_3.3) \n", "\n", "and if the flow is unconfined\n", "\n", "$$\n", "h = z_b + \\sqrt{2\\Phi/k} \\hskip 2em \\Phi\\le\\Phi_t\n", "$$(eq_headuc_3.3)" ] }, { "cell_type": "markdown", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "     As an example, the problem of {numref}`content:chap1:areal_recharge_two_rivers` is solved, but now the thickness of the aquifer is fixed to $H$ while the saturated thickness of the aquifer may vary with the head. The boundary conditions are\n", "\n", "$$\n", "\\Phi\\vert_{x=0}=\\Phi_l=kHh_l + C\n", "$$(eq_bc1_3.3)\n", "\n", "$$\n", "\\Phi\\vert_{x=L}=\\Phi_r=\\tfrac{1}{2}k(h_r-z_b)^2\n", "$$(eq_bc2_3.3) \n", "\n", "Application of the boundary conditions to the general solution {eq}`eq_secdif3.0` gives\n", "\n", "$$\n", "\\Phi = -\\frac{N}{2}(x^2 - Lx) + \\frac{(\\Phi_r - \\Phi_l)x}{L} + \\Phi_0\n", "$$(eq_general_3.3)\n", "\n", "The head solution is obtained from the potential with Eq. ({eq}`eq_headc_3.3`) or ({eq}`eq_headuc_3.3`). Results for the same parameters as {numref}`content:chap1:areal_recharge_two_rivers` are shown in {numref}`Figure %s `. Note that the flow is unconfined for only a small section of the aquifer near the right river." ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [], "source": [ "# parameters\n", "L = 1000 # aquifer length, m\n", "H = 10 # aquifer thickness, m\n", "zb = -5 # aquifer base, m\n", "k = 10 # hydraulic conductivity, m/d\n", "hl = 6 # specified head at the left boundary, m\n", "hr = 4 # specified head at the right boundary, m\n", "N = 0.001 # areal recharge, m/d" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [], "source": [ "# solution\n", "C = -0.5 * k * H**2 - k * H * zb\n", "phil = k * H * hl + C\n", "phir = 0.5 * k * (hr - zb)**2\n", "phit = 0.5 * k * H**2 # transition potential\n", "x = np.linspace(0, L, 400)\n", "phi = -N / 2 * (x ** 2 - L * x) + (phir - phil) * x / L + phil\n", "h = np.zeros_like(phi)\n", "h[phi >= phit] = (phi[phi > phit] - C) / (k * H)\n", "h[phi <= phit] = zb + np.sqrt(2 * phi[phi <= phit] / k)\n", "Qx = N * (x - L / 2) - (phir - phil) / L \n", "happrox = -N / (2 * k * H) * (x ** 2 - L * x) + (hr - hl) * x / L + hl\n", "Qxapprox = N * (x - L / 2) - k * H * (hr - hl) / L " ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "remove-output" ] }, "outputs": [ { "data": { "text/plain": [ "[]" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", " \n", " 2025-11-25T13:50:51.739065\n", " image/svg+xml\n", " \n", " \n", " Matplotlib v3.10.1, https://matplotlib.org/\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "\n" ], "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# basic plot\n", "plt.subplot(121)\n", "plt.plot(x, h, 'C0')\n", "plt.plot(x, happrox, 'C0--')\n", "plt.subplot(122)\n", "plt.plot(x, Qx, 'C1')\n", "plt.plot(x, Qxapprox, 'C1--')" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "remove-input", "remove-output" ] }, "outputs": [ { "data": { "application/papermill.record/image/svg+xml": "\n\n\n \n \n \n \n 2025-11-25T13:50:51.814713\n image/svg+xml\n \n \n Matplotlib v3.10.1, https://matplotlib.org/\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n", "application/papermill.record/text/plain": "
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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# plot\n", "fig = plt.figure(figsize=diagrams.figdef2)\n", "plt.subplot(121)\n", "plt.plot(x, h, label='variable T')\n", "plt.plot(x, happrox, 'C0--', label='constant T')\n", "plt.plot([0, L], [5, 5], color='k')\n", "plt.text(0, zb + H + 0.2, 'aquifer top')\n", "plt.plot([0, L], [zb, zb], color='k')\n", "plt.text(0, zb + 0.2, 'aquifer bottom')\n", "plt.xlabel('$x$ (m)')\n", "plt.ylabel('head (m)')\n", "plt.title('(a)', loc='left')\n", "plt.legend()\n", "plt.grid(True)\n", "plt.subplot(122)\n", "plt.plot(x, Qx, 'C1', label='variable T')\n", "plt.plot(x, Qxapprox, 'C1--', label='constant T')\n", "plt.xlabel('$x$ (m)')\n", "plt.ylabel('$Q_x$ (m$^2$/d)')\n", "plt.title('(b)', loc='left')\n", "plt.grid(True)\n", "plt.legend()\n", "plt.tight_layout()\n", "\n", "glue(\"fig_uncflow3_3\", fig, display=False);" ] }, { "cell_type": "markdown", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ " ```{glue:figure} fig_uncflow3_3 \n", "---\n", "name: fig_c_uncflow3.3\n", "---\n", "Combined confined/unconfined flow; head (left) and $Q_x$ (right).\n", "```" ] }, { "cell_type": "markdown", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "     The flux into the left river is now slightly higher than in {numref}`content:chap1:areal_recharge_two_rivers`, while the flux into the right river is slightly lower, so that the sum of the outflow in the two rivers is still equal to $NL$." ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "flux to left river: 0.305 m^2/d\n", "flux to right river: 0.695 m^2/d\n" ] } ], "source": [ "print(f'flux to left river: {-Qx[0]:.3f} m^2/d')\n", "print(f'flux to right river: {Qx[-1]:.3f} m^2/d')" ] }, { "cell_type": "markdown", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "     A contour plot of the steam function is created below ({numref}`Figure %s `). The top of the saturated zone is formed by the top of the aquifer in the zone where the flow is confined, and by the water table in the zone where the flow is unconfined." ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [], "source": [ "# solution\n", "psi = np.zeros((2, len(x)))\n", "psi[1] = -Qx\n", "xg = np.zeros_like(psi)\n", "xg[:] = x\n", "zg = np.zeros_like(psi)\n", "zg[0] = zb\n", "zg[1] = H + zb\n", "zg[1, h < H + zb] = h[h < H + zb]" ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "remove-output" ] }, "outputs": [ { "data": { "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", " \n", " 2025-11-25T13:50:51.967056\n", " image/svg+xml\n", " \n", " \n", " Matplotlib v3.10.1, https://matplotlib.org/\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "\n" ], "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# basic streamline plot\n", "plt.subplot(111, aspect=25)\n", "plt.contour(xg, zg, psi, 20, colors='C1', linestyles='-')\n", "plt.plot(xg[0], zg[1], 'C0');" ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "remove-input", "remove-output" ] }, "outputs": [ { "data": { "application/papermill.record/image/svg+xml": "\n\n\n \n \n \n \n 2025-11-25T13:50:52.010163\n image/svg+xml\n \n \n Matplotlib v3.10.1, https://matplotlib.org/\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n", "application/papermill.record/text/plain": "
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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# plot\n", "fig = plt.figure(figsize=diagrams.figdef2)\n", "aspect = 25\n", "plt.subplot(111, aspect=aspect)\n", "cs = plt.contour(xg, zg, psi, 0.1 * np.arange(-7, 3, 1), colors='C1', linestyles='-')\n", "plt.clabel(cs, fmt='%1.2f', fontsize='smaller')\n", "plt.contour(xg, zg, psi, np.arange(-0.65, 0.3, 0.1), colors='C1', linestyles='-')\n", "plt.contour(xg, zg, psi, [0], colors='C1')\n", "plt.plot(xg[0], zg[1], 'C0')\n", "plt.xlabel('$x$ (m)')\n", "plt.ylabel(f'$z$ (m) - VE:{aspect}x')\n", "plt.yticks(np.linspace(-5, 5, 5))\n", "plt.grid()\n", "plt.tight_layout()\n", "\n", "glue(\"fig_stream3_3\", fig, display=False);" ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "remove-input", "remove-output" ] }, "outputs": [ { "data": { "application/javascript": [ "'use strict';\n", "(function(root) {\n", " function now() {\n", " return new Date();\n", " }\n", "\n", " const force = true;\n", "\n", " if (typeof root._bokeh_onload_callbacks === \"undefined\" || force === true) {\n", " root._bokeh_onload_callbacks = [];\n", " root._bokeh_is_loading = undefined;\n", " }\n", "\n", "const JS_MIME_TYPE = 'application/javascript';\n", " const HTML_MIME_TYPE = 'text/html';\n", " const EXEC_MIME_TYPE = 'application/vnd.bokehjs_exec.v0+json';\n", " const CLASS_NAME = 'output_bokeh rendered_html';\n", "\n", " /**\n", " * Render data to the DOM node\n", " */\n", " function render(props, node) {\n", " const script = document.createElement(\"script\");\n", " node.appendChild(script);\n", " }\n", "\n", " /**\n", " * Handle when an output is cleared or removed\n", " */\n", " function handleClearOutput(event, handle) {\n", " function drop(id) {\n", " const view = Bokeh.index.get_by_id(id)\n", " if (view != null) {\n", " view.model.document.clear()\n", " Bokeh.index.delete(view)\n", " }\n", " }\n", "\n", " const cell = handle.cell;\n", "\n", " const id = cell.output_area._bokeh_element_id;\n", " const server_id = cell.output_area._bokeh_server_id;\n", "\n", " // Clean up Bokeh references\n", " if (id != null) {\n", " drop(id)\n", " }\n", "\n", " if (server_id !== undefined) {\n", " // Clean up Bokeh references\n", " const cmd_clean = \"from bokeh.io.state import curstate; print(curstate().uuid_to_server['\" + server_id + \"'].get_sessions()[0].document.roots[0]._id)\";\n", " cell.notebook.kernel.execute(cmd_clean, {\n", " iopub: {\n", " output: function(msg) {\n", " const id = msg.content.text.trim()\n", " drop(id)\n", " }\n", " }\n", " });\n", " // Destroy server and session\n", " const cmd_destroy = \"import bokeh.io.notebook as ion; ion.destroy_server('\" + server_id + \"')\";\n", " cell.notebook.kernel.execute(cmd_destroy);\n", " }\n", " }\n", "\n", " /**\n", " * Handle when a new output is added\n", " */\n", " function handleAddOutput(event, handle) {\n", " const output_area = handle.output_area;\n", " const output = handle.output;\n", "\n", " // limit handleAddOutput to display_data with EXEC_MIME_TYPE content only\n", " if ((output.output_type != \"display_data\") || (!Object.prototype.hasOwnProperty.call(output.data, EXEC_MIME_TYPE))) {\n", " return\n", " }\n", "\n", " const toinsert = output_area.element.find(\".\" + CLASS_NAME.split(' ')[0]);\n", "\n", " if (output.metadata[EXEC_MIME_TYPE][\"id\"] !== undefined) {\n", " toinsert[toinsert.length - 1].firstChild.textContent = output.data[JS_MIME_TYPE];\n", " // store reference to embed id on output_area\n", " output_area._bokeh_element_id = output.metadata[EXEC_MIME_TYPE][\"id\"];\n", " }\n", " if (output.metadata[EXEC_MIME_TYPE][\"server_id\"] !== undefined) {\n", " const bk_div = document.createElement(\"div\");\n", " bk_div.innerHTML = output.data[HTML_MIME_TYPE];\n", " const script_attrs = bk_div.children[0].attributes;\n", " for (let i = 0; i < script_attrs.length; i++) {\n", " toinsert[toinsert.length - 1].firstChild.setAttribute(script_attrs[i].name, script_attrs[i].value);\n", " toinsert[toinsert.length - 1].firstChild.textContent = bk_div.children[0].textContent\n", " }\n", " // store reference to server id on output_area\n", " output_area._bokeh_server_id = output.metadata[EXEC_MIME_TYPE][\"server_id\"];\n", " }\n", " }\n", "\n", " function register_renderer(events, OutputArea) {\n", "\n", " function append_mime(data, metadata, element) {\n", " // create a DOM node to render to\n", " const toinsert = this.create_output_subarea(\n", " metadata,\n", " CLASS_NAME,\n", " EXEC_MIME_TYPE\n", " );\n", " this.keyboard_manager.register_events(toinsert);\n", " // Render to node\n", " const props = {data: data, metadata: metadata[EXEC_MIME_TYPE]};\n", " render(props, toinsert[toinsert.length - 1]);\n", " element.append(toinsert);\n", " return toinsert\n", " }\n", "\n", " /* Handle when an output is cleared or removed */\n", " events.on('clear_output.CodeCell', handleClearOutput);\n", " events.on('delete.Cell', handleClearOutput);\n", "\n", " /* Handle when a new output is added */\n", " events.on('output_added.OutputArea', handleAddOutput);\n", "\n", " /**\n", " * Register the mime type and append_mime function with output_area\n", " */\n", " OutputArea.prototype.register_mime_type(EXEC_MIME_TYPE, append_mime, {\n", " /* Is output safe? */\n", " safe: true,\n", " /* Index of renderer in `output_area.display_order` */\n", " index: 0\n", " });\n", " }\n", "\n", " // register the mime type if in Jupyter Notebook environment and previously unregistered\n", " if (root.Jupyter !== undefined) {\n", " const events = require('base/js/events');\n", " const OutputArea = require('notebook/js/outputarea').OutputArea;\n", "\n", " if (OutputArea.prototype.mime_types().indexOf(EXEC_MIME_TYPE) == -1) {\n", " register_renderer(events, OutputArea);\n", " }\n", " }\n", " if (typeof (root._bokeh_timeout) === \"undefined\" || force === true) {\n", " root._bokeh_timeout = Date.now() + 5000;\n", " root._bokeh_failed_load = false;\n", " }\n", "\n", " const NB_LOAD_WARNING = {'data': {'text/html':\n", " \"
\\n\"+\n", " \"

\\n\"+\n", " \"BokehJS does not appear to have successfully loaded. If loading BokehJS from CDN, this \\n\"+\n", " \"may be due to a slow or bad network connection. Possible fixes:\\n\"+\n", " \"

\\n\"+\n", " \"
    \\n\"+\n", " \"
  • re-rerun `output_notebook()` to attempt to load from CDN again, or
    • \\n\"+\n", " \"
  • use INLINE resources instead, as so:
    • \\n\"+\n", " \"
\\n\"+\n", " \"\\n\"+\n", " \"from bokeh.resources import INLINE\\n\"+\n", " \"output_notebook(resources=INLINE)\\n\"+\n", " \"\\n\"+\n", " \"
\"}};\n", "\n", " function display_loaded(error = null) {\n", " const el = document.getElementById(null);\n", " if (el != null) {\n", " const html = (() => {\n", " if (typeof root.Bokeh === \"undefined\") {\n", " if (error == null) {\n", " return \"BokehJS is loading ...\";\n", " } else {\n", " return \"BokehJS failed to load.\";\n", " }\n", " } else {\n", " const prefix = `BokehJS ${root.Bokeh.version}`;\n", " if (error == null) {\n", " return `${prefix} successfully loaded.`;\n", " } else {\n", " return `${prefix} encountered errors while loading and may not function as expected.`;\n", " }\n", " }\n", " })();\n", " el.innerHTML = html;\n", "\n", " if (error != null) {\n", " const wrapper = document.createElement(\"div\");\n", " wrapper.style.overflow = \"auto\";\n", " wrapper.style.height = \"5em\";\n", " wrapper.style.resize = \"vertical\";\n", " const content = document.createElement(\"div\");\n", " content.style.fontFamily = \"monospace\";\n", " content.style.whiteSpace = \"pre-wrap\";\n", " content.style.backgroundColor = \"rgb(255, 221, 221)\";\n", " content.textContent = error.stack ?? error.toString();\n", " wrapper.append(content);\n", " el.append(wrapper);\n", " }\n", " } else if (Date.now() < root._bokeh_timeout) {\n", " setTimeout(() => display_loaded(error), 100);\n", " }\n", " }\n", "\n", " function run_callbacks() {\n", " try {\n", " root._bokeh_onload_callbacks.forEach(function(callback) {\n", " if (callback != null)\n", " callback();\n", " });\n", " } finally {\n", " delete root._bokeh_onload_callbacks\n", " }\n", " console.debug(\"Bokeh: all callbacks have finished\");\n", " }\n", "\n", " function load_libs(css_urls, js_urls, callback) {\n", " if (css_urls == null) css_urls = [];\n", " if (js_urls == null) js_urls = [];\n", "\n", " root._bokeh_onload_callbacks.push(callback);\n", " if (root._bokeh_is_loading > 0) {\n", " console.debug(\"Bokeh: BokehJS is being loaded, scheduling callback at\", now());\n", " return null;\n", " }\n", " if (js_urls == null || js_urls.length === 0) {\n", " run_callbacks();\n", " return null;\n", " }\n", " console.debug(\"Bokeh: BokehJS not loaded, scheduling load and callback at\", now());\n", " root._bokeh_is_loading = css_urls.length + js_urls.length;\n", "\n", " function on_load() {\n", " root._bokeh_is_loading--;\n", " if (root._bokeh_is_loading === 0) {\n", " console.debug(\"Bokeh: all BokehJS libraries/stylesheets loaded\");\n", " run_callbacks()\n", " }\n", " }\n", "\n", " function on_error(url) {\n", " console.error(\"failed to load \" + url);\n", " }\n", "\n", " for (let i = 0; i < css_urls.length; i++) {\n", " const url = css_urls[i];\n", " const element = document.createElement(\"link\");\n", " element.onload = on_load;\n", " element.onerror = on_error.bind(null, url);\n", " element.rel = \"stylesheet\";\n", " element.type = \"text/css\";\n", " element.href = url;\n", " console.debug(\"Bokeh: injecting link tag for BokehJS stylesheet: \", url);\n", " document.body.appendChild(element);\n", " }\n", "\n", " for (let i = 0; i < js_urls.length; i++) {\n", " const url = js_urls[i];\n", " const element = document.createElement('script');\n", " element.onload = on_load;\n", " element.onerror = on_error.bind(null, url);\n", " element.async = false;\n", " element.src = url;\n", " console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n", " document.head.appendChild(element);\n", " }\n", " };\n", "\n", " function inject_raw_css(css) {\n", " const element = document.createElement(\"style\");\n", " element.appendChild(document.createTextNode(css));\n", " document.body.appendChild(element);\n", " }\n", "\n", " const js_urls = [\"https://cdn.bokeh.org/bokeh/release/bokeh-3.7.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-gl-3.7.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-3.7.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-3.7.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-mathjax-3.7.3.min.js\"];\n", " const css_urls = [];\n", "\n", " const inline_js = [ function(Bokeh) {\n", " Bokeh.set_log_level(\"info\");\n", " },\n", "function(Bokeh) {\n", " }\n", " ];\n", "\n", " function run_inline_js() {\n", " if (root.Bokeh !== undefined || force === true) {\n", " try {\n", " for (let i = 0; i < inline_js.length; i++) {\n", " inline_js[i].call(root, root.Bokeh);\n", " }\n", "\n", " } catch (error) {throw error;\n", " }} else if (Date.now() < root._bokeh_timeout) {\n", " setTimeout(run_inline_js, 100);\n", " } else if (!root._bokeh_failed_load) {\n", " console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n", " root._bokeh_failed_load = true;\n", " } else if (force !== true) {\n", " const cell = $(document.getElementById(null)).parents('.cell').data().cell;\n", " cell.output_area.append_execute_result(NB_LOAD_WARNING)\n", " }\n", " }\n", "\n", " if (root._bokeh_is_loading === 0) {\n", " console.debug(\"Bokeh: BokehJS loaded, going straight to plotting\");\n", " run_inline_js();\n", " } else {\n", " load_libs(css_urls, js_urls, function() {\n", " console.debug(\"Bokeh: BokehJS plotting callback run at\", now());\n", " run_inline_js();\n", " });\n", " }\n", "}(window));" ], "application/vnd.bokehjs_load.v0+json": "'use strict';\n(function(root) {\n function now() {\n return new Date();\n }\n\n const force = true;\n\n if (typeof root._bokeh_onload_callbacks === \"undefined\" || force === true) {\n root._bokeh_onload_callbacks = [];\n root._bokeh_is_loading = undefined;\n }\n\n\n if (typeof (root._bokeh_timeout) === \"undefined\" || force === true) {\n root._bokeh_timeout = Date.now() + 5000;\n root._bokeh_failed_load = false;\n }\n\n const NB_LOAD_WARNING = {'data': {'text/html':\n \"
\\n\"+\n \"

\\n\"+\n \"BokehJS does not appear to have successfully loaded. If loading BokehJS from CDN, this \\n\"+\n \"may be due to a slow or bad network connection. Possible fixes:\\n\"+\n \"

\\n\"+\n \"
    \\n\"+\n \"
  • re-rerun `output_notebook()` to attempt to load from CDN again, or
    • \\n\"+\n \"
  • use INLINE resources instead, as so:
    • \\n\"+\n \"
\\n\"+\n \"\\n\"+\n \"from bokeh.resources import INLINE\\n\"+\n \"output_notebook(resources=INLINE)\\n\"+\n \"\\n\"+\n \"
\"}};\n\n function display_loaded(error = null) {\n const el = document.getElementById(null);\n if (el != null) {\n const html = (() => {\n if (typeof root.Bokeh === \"undefined\") {\n if (error == null) {\n return \"BokehJS is loading ...\";\n } else {\n return \"BokehJS failed to load.\";\n }\n } else {\n const prefix = `BokehJS ${root.Bokeh.version}`;\n if (error == null) {\n return `${prefix} successfully loaded.`;\n } else {\n return `${prefix} encountered errors while loading and may not function as expected.`;\n }\n }\n })();\n el.innerHTML = html;\n\n if (error != null) {\n const wrapper = document.createElement(\"div\");\n wrapper.style.overflow = \"auto\";\n wrapper.style.height = \"5em\";\n wrapper.style.resize = \"vertical\";\n const content = document.createElement(\"div\");\n content.style.fontFamily = \"monospace\";\n content.style.whiteSpace = \"pre-wrap\";\n content.style.backgroundColor = \"rgb(255, 221, 221)\";\n content.textContent = error.stack ?? error.toString();\n wrapper.append(content);\n el.append(wrapper);\n }\n } else if (Date.now() < root._bokeh_timeout) {\n setTimeout(() => display_loaded(error), 100);\n }\n }\n\n function run_callbacks() {\n try {\n root._bokeh_onload_callbacks.forEach(function(callback) {\n if (callback != null)\n callback();\n });\n } finally {\n delete root._bokeh_onload_callbacks\n }\n console.debug(\"Bokeh: all callbacks have finished\");\n }\n\n function load_libs(css_urls, js_urls, callback) {\n if (css_urls == null) css_urls = [];\n if (js_urls == null) js_urls = [];\n\n root._bokeh_onload_callbacks.push(callback);\n if (root._bokeh_is_loading > 0) {\n console.debug(\"Bokeh: BokehJS is being loaded, scheduling callback at\", now());\n return null;\n }\n if (js_urls == null || js_urls.length === 0) {\n run_callbacks();\n return null;\n }\n console.debug(\"Bokeh: BokehJS not loaded, scheduling load and callback at\", now());\n root._bokeh_is_loading = css_urls.length + js_urls.length;\n\n function on_load() {\n root._bokeh_is_loading--;\n if (root._bokeh_is_loading === 0) {\n console.debug(\"Bokeh: all BokehJS libraries/stylesheets loaded\");\n run_callbacks()\n }\n }\n\n function on_error(url) {\n console.error(\"failed to load \" + url);\n }\n\n for (let i = 0; i < css_urls.length; i++) {\n const url = css_urls[i];\n const element = document.createElement(\"link\");\n element.onload = on_load;\n element.onerror = on_error.bind(null, url);\n element.rel = \"stylesheet\";\n element.type = \"text/css\";\n element.href = url;\n console.debug(\"Bokeh: injecting link tag for BokehJS stylesheet: \", url);\n document.body.appendChild(element);\n }\n\n for (let i = 0; i < js_urls.length; i++) {\n const url = js_urls[i];\n const element = document.createElement('script');\n element.onload = on_load;\n element.onerror = on_error.bind(null, url);\n element.async = false;\n element.src = url;\n console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n document.head.appendChild(element);\n }\n };\n\n function inject_raw_css(css) {\n const element = document.createElement(\"style\");\n element.appendChild(document.createTextNode(css));\n document.body.appendChild(element);\n }\n\n const js_urls = [\"https://cdn.bokeh.org/bokeh/release/bokeh-3.7.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-gl-3.7.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-3.7.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-3.7.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-mathjax-3.7.3.min.js\"];\n const css_urls = [];\n\n const inline_js = [ function(Bokeh) {\n Bokeh.set_log_level(\"info\");\n },\nfunction(Bokeh) {\n }\n ];\n\n function run_inline_js() {\n if (root.Bokeh !== undefined || force === true) {\n try {\n for (let i = 0; i < inline_js.length; i++) {\n inline_js[i].call(root, root.Bokeh);\n }\n\n } catch (error) {throw error;\n }} else if (Date.now() < root._bokeh_timeout) {\n setTimeout(run_inline_js, 100);\n } else if (!root._bokeh_failed_load) {\n console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n root._bokeh_failed_load = true;\n } else if (force !== true) {\n const cell = $(document.getElementById(null)).parents('.cell').data().cell;\n cell.output_area.append_execute_result(NB_LOAD_WARNING)\n }\n }\n\n if (root._bokeh_is_loading === 0) {\n console.debug(\"Bokeh: BokehJS loaded, going straight to plotting\");\n run_inline_js();\n } else {\n load_libs(css_urls, js_urls, function() {\n console.debug(\"Bokeh: BokehJS plotting callback run at\", now());\n run_inline_js();\n });\n }\n}(window));" }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "\n", "
\n" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "application/javascript": [ "(function(root) {\n", " function embed_document(root) {\n", " const docs_json = {\"3760c21c-4c09-4ebb-8f9e-69822082e72e\":{\"version\":\"3.7.3\",\"title\":\"Bokeh Application\",\"roots\":[{\"type\":\"object\",\"name\":\"Column\",\"id\":\"p1392\",\"attributes\":{\"children\":[{\"type\":\"object\",\"name\":\"Div\",\"id\":\"p1391\",\"attributes\":{\"styles\":{\"type\":\"map\",\"entries\":[[\"margin\",\"0 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const h0 = 6;\\n const k = 10;\\n const H = 10;\\n const zb = -5;\\n const L = 1000;\\n const N = 0.001;\\n const levels = Array.from({length: 28}, (_, i) => -0.9 + 0.1 * i);\\n \\n const hL = slider.value;\\n const C = -0.5 * k * H * H - k * H * zb;\\n const phi0 = k * H * h0 + C;\\n const phi1 = 0.5 * k * Math.pow(hL - zb, 2);\\n const phit = 0.5 * k * H * H;\\n \\n const x = Array.from({length: 400}, (_, i) => i * L / 399);\\n const phi = x.map(xi => -N / 2 * (xi * xi - L * xi) + (phi1 - phi0) * xi / L + phi0);\\n \\n let h = [];\\n for (let i = 0; i < phi.length; i++) {\\n if (phi[i] >= phit) {\\n h.push((phi[i] - C) / (k * H));\\n } else {\\n h.push(zb + Math.sqrt(2 * phi[i] / k));\\n }\\n }\\n \\n // Update watertable line (blue)\\n const zg = [];\\n for (let i = 0; i < h.length; i++) {\\n if (h[i] < H + zb) {\\n zg.push(h[i]);\\n } else {\\n zg.push(H + zb);\\n }\\n }\\n source_grid.data['zg'] = zg;\\n source_grid.change.emit();\\n \\n // Update contour lines\\n let idx = slider_vals.findIndex(v => Math.abs(v - hL) < 1e-6);\\n if (idx === -1) return;\\n \\n const xs = all_xs[idx];\\n const ys = all_ys[idx];\\n \\n for (let i = 0; i < levels.length; i++) {\\n sources[i].data['x_values'] = xs[i];\\n sources[i].data['y_values'] = ys[i];\\n sources[i].change.emit();\\n } \\n \"}}]]]},\"width\":570,\"margin\":10,\"align\":\"center\",\"title\":\"h\\u1d63 (m)\",\"start\":0,\"end\":5,\"value\":4,\"step\":0.25}},{\"type\":\"object\",\"name\":\"Div\",\"id\":\"p1389\",\"attributes\":{\"align\":\"center\",\"text\":\"
0 m              |              1 m              |              2 m              |              3 m              |              4 m              |              5 m
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\n", " Solution \n", "\n", "$h=6.550$ m for $H=10$ m\\\n", "$h=6.446$ m for $H=20$ m\n", "\n", "\n", "\n", "
\n", "\n", "```" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "celltoolbar": "Tags", "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.13.7" }, "latex_envs": { "LaTeX_envs_menu_present": true, "autoclose": false, "autocomplete": true, "bibliofile": "biblio.bib", "cite_by": "apalike", "current_citInitial": 1, "eqLabelWithNumbers": true, "eqNumInitial": 1, "hotkeys": { "equation": "Ctrl-E", "itemize": "Ctrl-I" }, "labels_anchors": false, "latex_user_defs": false, "report_style_numbering": false, "user_envs_cfg": false }, "varInspector": { "cols": { "lenName": 16, "lenType": 16, "lenVar": 40 }, "kernels_config": { "python": { "delete_cmd_postfix": "", "delete_cmd_prefix": "del ", "library": "var_list.py", "varRefreshCmd": "print(var_dic_list())" }, "r": { "delete_cmd_postfix": ") ", "delete_cmd_prefix": "rm(", "library": "var_list.r", "varRefreshCmd": "cat(var_dic_list()) " } }, "types_to_exclude": [ "module", "function", "builtin_function_or_method", "instance", "_Feature" ], "window_display": false } }, "nbformat": 4, "nbformat_minor": 4 }