If the current page has child menu items, display them. Hazzah!

menus.php

<?php
/**
 * Returns the submenu items of the parent menu item.
 * @param $sorted_menu_items
 * @param $args
 * @return mixed
 */
function theme_wp_nav_menu_sub_menu_objects( $sorted_menu_items, $args ) {
    if ( isset( $args->sub_menu ) ) {
        $root_id = 0;

        // find the current menu item
        foreach ( $sorted_menu_items as $menu_item ) {
            if ( $menu_item->current ) {
                // set the root id based on whether the current menu item has a parent or not
                $root_id = ( $menu_item->menu_item_parent ) ? $menu_item->menu_item_parent : $menu_item->ID;
                break;
            }
        }

        $menu_item_parents = array();
        foreach ( $sorted_menu_items as $key => $item ) {
            // init menu_item_parents
            if ( $item->ID == $root_id ) $menu_item_parents[] = $item->ID;

            if ( in_array( $item->menu_item_parent, $menu_item_parents ) ) {
                // part of sub-tree: keep!
                $menu_item_parents[] = $item->ID;
            } else {
                // not part of sub-tree: away with it!
                unset( $sorted_menu_items[$key] );
            }
        }

        return $sorted_menu_items;

    } else {
        return $sorted_menu_items;
    }
}

add_filter( 'wp_nav_menu_objects', 'theme_wp_nav_menu_sub_menu_objects', 10, 2 );

function theme_primary_menu() {
    $primary = wp_nav_menu( array(
        'theme_location'  => 'primary',
        'container'       => '',
        'container_class' => '',
        'echo'            => false,
        'fallback_cb'     => '',
        'items_wrap'      => '<ul id="%1$s" class="%2$s">%3$s</ul>',
        'depth'           => 0,
    ));

    echo $primary;
}
    
function theme_subnav_menu() {
    $subnav = wp_nav_menu( array(
        'theme_location'  => 'primary',
        'container'       => '',
        'container_class' => '',
        'fallback_cb'     => '',
        'items_wrap'      => '<ul id="%1$s" class="%2$s sub-menu">%3$s</ul>',
        'echo'            => false,
        'sub_menu'        => true
    ) );

    $menu_items = substr_count( $subnav, 'class="menu-item ' );

    if ( $menu_items != 0 ) {
        echo $subnav;
    }

}